Texas Go Math

Refer to our Texas Go Math Grade 8 Answer Key Pdf to score good marks in the exams. Test yourself by practicing the problems from Texas Go Math Grade 8 Lesson 6.3 Answer Key Comparing Functions.

Texas Go Math Grade 8 Lesson 6.3 Answer Key Comparing Functions

Essential Question
How can you use tables, graphs, and equations to compare functions?

Your Turn

Question 1.
Quentin is choosing between buying books at the bookstore or buying online versions of the books for his tablet. The cost, y dollars, of ordering books online for x books is y = 6.95x + 1.50. The cost of buying the books at the bookstore is shown in the table. Which method of buying books is more expensive if Quentin wants to buy 6 books?

Answer:
y = 6.95x + 1.5 Buying the books online
y = 6.95(6) + 1.5 = $43.2 When Quentin wants to buy 6 books, substitute x = 6
Slope = \(\frac{15-7.5}{2-1}\) = \(\frac{7.5}{1}\) = 7.5 Writing the equation for buying the books at the bookstore. Find the slope using given points by Slope (m) = \(\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\) where (x₂, y₂) = (2, 15) and (x₁, y₁) = (1, 7.5)
15 = 7.5(2) + b Substituting the value of slope (m) and (x, y) in the
15 = 15 + b slope intercept form to find y intercept (b): y = mx + b
b = 15 – 15 = 0
y = 7.5x Substituting the value of slope (m) and y intercept (b) in slope intercept form: y = mx + b
y = 7.5(6) = $45 When Quentin wants to buy 6 books, substitute x = 6
Buying books at the bookstore is more expensive. Compare the cost of buying 6 books $43.2 < $45

Texas Go Math Grade 8 Lesson 6.3 Explore Activity Answer Key

Explore Activity 1

Comparing a Table and a Graph
The table and graph show how many words Morgan and Brian typed correctly on a typing test. For both students, the relationship between words typed correctly and time is linear.

A. Find Morgan’s unit rate.

B. Find Brian’s unit rate.

C. Which student types more correct words per minute?

Reflect

Lesson 6.3 Comparing Functions Answer Key Go Math 8th Grade Question 2.
Katie types 17 correct words per minute. Explain how a graph of Katie’s test results would compare to Morgan’s and Brian’s.
Answer:
Katie types 17 correct words per minute; Brian types 20 correct words per minute and Morgan types 15 correct words per minute. Therefore, the graph of Katie’s test results will be a straight line that passes through the origin (0, 0). The value of its slope is greater than the slope of Morgan’s graph of test results and smaller than the slope of Brian’s graph of test results. So, the graph will be a straight line that lies between the line of Morgan and Brian.

Explore Activity 2
Comparing a Graph and a Description
Jamal wants to buy a new game system that costs $200. He does not have enough money to buy it today, so he compares layaway plans at different stores.

The plan at Store A is shown on the graph.
Store B requires an initial payment of $60 and weekly payments of $20 until the balance is paid in full.

A. Write an equation in slope-intercept form for Store A’s layaway plan. Let x represent the number of weeks and y represent the balance owed.

B. Write an equation in slope-intercept form for Store B’s layaway plan. Let x represent the number of weeks and y represent the balance owed.

C. Sketch a graph of the plan at Store B on the same grid as Store A.

D. How can you use the graphs to tell which plan requires the greater down payment? How can you use the equations?

E. How can you Use the graphs to tell which plan requires the greater weekly payment? How can you use the equations?

F. Which plan allows Jamal to pay for the game system faster? Explain.

Texas Go Math Grade 8 Lesson 6.3 Guided Practice Answer Key

Doctors have two methods of calculating maximum heart rate. With the first method, the maximum heart rate,y, in beats per minute is y = 220 – x, where x is the person’s age. The maximum heart rate with the second method is shown in the table. (Example 1)

Question 1.
Which method gives the greater maximum heart rate for a 70-year-old?
Answer:
y = 220 – x First method
y = 220 – 70 = 150 Substitute x = 70 yrs
slope = \(\frac{187-194}{30-20}\) = \(\frac{-7}{10}\) = -0.7
Writing the equation for the second method. Find the slope using given points by slope (m) = \(\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\) where (x₂, y₂) = (30, 187) and (x₁, y₁) = (20, 194)
194 = -0.7(20) + b Substituting the value of slope (m) and (x, y) in the slope-intercept form to find y-intercept (b): y = mx + b
194 = -14 + b
b = 194 + 14 = 208

y = 208 – 0.7x Substituting the value of slope (m) and y-intercept (b) in slope-intercept form: y = mx + b
y = 208 – 0.7(70) = 159 Substitute x = 70 yrs
The second method gives the greater maximum heart rate for a 70-year-old methods

Compare the maximum heart rate for the two methods
150 < 159

Question 2.
Are heart rate and age proportional or nonproportional for each method?
Answer:
For method 1, the relationship is non-proportional. Comparing with the general linear form of an equation: y = mx + b.
For method 2, the relationship is non-proportional. Since b ≠ 0, the relationship is not proportional.

Aisha runs a tutoring business. With Plan 1, students may choose to pay $15 per hour. With Plan 2, they may follow the plan shown on the graph. (Explore Activity 1 and 2)

Question 3.
Describe the plan shown on the graph.
Answer:
We choose two points on the graph to find the slope. We substitute (0, 40) for (x₁, y₁) and (4, 60) for (x₂, y₂).
m = \(\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\) = \(\frac{60-40}{4-0}\) = \(\frac{20}{4}\) = 5
We read the y-intercept from the graph, which is 40.
b = 40
We use the slope and y-intercept values that we found to write an equation in slope-intercept form.
y = mx + b
y = 5x + 40
This means that
Plan 2 has an initial cost of $40 and a rate of $5 per hour

Question 4.
Sketch a graph of Plan 1 on the same grid as Plan 2.
Answer:

Plan 1 is represented by red and Plan 2 is represented by blue

8th Grade Comparing Functions Answer Key Lesson 6.3 Question 5.
What does the intersection of the two graphs mean?
Answer:
The intersection of the two graphs represents the number of hours for which both plans will cost the same.

Question 6.
Which plan is cheaper for 10 hours of tutoring?
Answer:
Knowing that with Plan 1 student pays 15 dollars per hour, we can calculate the cost of 10 hours of tutoring. Variable y is the cost and variable æ is the number of hours. Therefore:
y₁ = 15x
y₁ = 15 . 10
y₁ =150

To calculate the cost of Plan 2, we will use the following formula:
y₂ = 5x + 40,
where variable y is the cost and variable x is the number of hours. Therefore:
y₂ = 5 . 10 + 40
y₂ = 90
Hence, we can conclude that Plan 2 is cheaper for 10 hours of tutoring.

Question 7.
Are cost and time proportional or nonproportional for each plan?
Answer:
The cost and time are proportional for Plan 1 Compared with the general linear form of equation: y = mx + b. Since b = 0, the relationship is proportional.

The cost and time are not proportional for Plan 2 Compared with the general linear form of equation: y = mx + b.
Since b ≠ 0, the relationship is not proportional.

Essential Question Check-In

Go Math Grade 8 Lesson 6.3 Answer Key Question 8.
When using tables, graphs, and equations to compare functions, why do you find the equations for tables and graphs?
Answer:
The tables and graphs represent a part of the solution of the function By writing the equation, any value can be substituted to evaluate the function and compare it with the equations.

Texas Go Math Grade 8 Lesson 6.3 Independent Practice Answer Key

The table and graph show the miles driven and gas used for two scooters.

Question 9.
Which scooter uses fewer gallons of gas when 1350 miles are driven?
Answer:
Looking at the given table for scooter A, we can conclude that scooter uses 2 gallons for every 150 miles. Therefore, we can write the equation where variable y represents used gallons of gas and variable x represents driven miles.
y_a = \(\frac{2 x}{150}\)
y_a = \(\frac{x}{75}\)
Considering the fact that we are testing the consumption of gas for 1350 driven miles, we can use the equation for y_a and insert x = 1350 as follows:
y_a = \(\frac{1350}{75}\)
y_a = 18
Given graph represents driven miles and gas used for scooter B. We will find value of driven miles for gas
consumption of 1 gallon by drawing a horizontal line on the value of 1 gallon and looking for an intersection with the
function. Lines intersect on x = 90, meaning that scooter B uses 1 gallon of gas for 90 driven miles. Therefore:
y_b = \(\frac{x}{90}\),
where y_b is the consumption of gas and variable x represents driven miles.

Using the equation obtained in the previous step, we can calculate gas consumption for 1350 driven miles.
y_b = \(\frac{1350}{90}\)
y_b = 15
Hence, we can conclude that scooter B uses fewer gallons of gas when driving 1350 miles.

Lesson 6.3 Answer Key 8th Grade Comparing Functions Question 10.
Are gas used and miles proportional or nonproportional for each scooter?
Answer:
The gas used and miles are proportional for both scooters.

Comparing with the general linear form of an equation: y = mx + b. Since b = 0, the relationship is proportional.

A cell phone company offers two texting plans to its customers. The monthly cost, y dollars, of one plan, is y = 0.10x + 5, where x is the number of texts. The cost of the other plan is shown in the table.

Question 11.
Which plan is cheaper for under 200 texts? ______
Answer:
y = 0.10x + 5 Plan 1
y = 0.10(199) + 5 = $24.90 Substitute x = 199
slope = \(\frac{25-20}{200-100}\) = \(\frac{5}{100}\) = 0.05 Writing the equation for plan 2. Find the slope using given points by Slope (m) = \(\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\)
where (x₂, y₂) = (200, 25) (x₁, y₁) = (100, 20)
20 = 0.05(100) + b Substituting the value of slope (m) and (x, y) in the slope-intercept form to find y-intercept (b):
20 = 5 + b y = mx + b
b = 20 – 5 = 15
y = 0.05x + 15 Substituting the value of slope (m) and y-intercept (b) in slope-intercept form: y = mx + b
y = 0.05(199) + 15 = $24.95 Substitute x = 199

Plan 1 is cheaper Compare the cost for two plans for text < 200
$24.90 < $24.95

Question 12.
The graph of the first plan does not pass through the origin. What does this indicate?
Answer:
y = 0.10x + 5 Plan 1
The graph that does not pass through the origin indicates that there is a base price of $\$5$ for the plan.

Go Math 8th Grade Lesson 6.3 Comparing Functions Question 13.
Brianna wants to buy a digital camera for a photography class. One store offers the camera for $50 down and a payment plan of $20 per month. The payment plan for a second store is described by y = 15x + 80, where y is the total cost in dollars and x is the number of months. Which camera is cheaper when the camera is paid off in 12 months? Explain.
Answer:
For us to determine which camera is cheaper when the camera is paid off in 12 months, let us determine the total cost of the digital camera offered by the two stores.

The first store offers the camera for $50 down and a payment of $20 per month. By applying the slope-intercept form y = mx + b. Let y be the total cost of the camera for the first store x be the number of months the camera needs to be paid off m be the payment every month b be the down payment

Substituting the values of m = 20, b = 50, and x = 12, the total cost of the camera offered by the first store is
\begin{aligned}
y&=mx+b\
y&=20(12)+50\
y&=240+50\
y&=290
\end{aligned}

Thus, the total cost of the camera offered by the first store is $290

The payment plan for the second store is described by y = 15x + 80
where y is the total. cost of the camera in dollars and x is the number of months

Thus, the total cost of the camera offered by the second store is $260

Now, let us compare the total costs of the camera offered by the two stores. The total cost of the camera offered by the first store is $290 while the second store camera’s total cost is $260. Since $260 is less than $290, then we can conclude that **the camera of the second store is cheaper than the camera of the first store**.

Question 14.
The French club and soccer team are washing cars to earn money. The amount earned, y dollars, for washing x cars is a linear function. Which group makes the most money per car? Explain.

Answer:
slope = \(\frac{20-10}{4-2}\) = \(\frac{10}{2}\) = 5 Find the slope/rate of change for the French club. Find the slope using the given points by
Slope (m) = \(\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\)
French club makes $\$5$ per car
where
(x₂, y₂) = (4, 20) and (x₁, y₁) = (2, 10)
slope = \(\frac{16-0}{2-0}\) = \(\frac{16}{2}\) = 8 Find the slope/rate of change for a soccer club. Find the slope using the given points by
Slope (m) = \(\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\) where
(x₂, y₂) = (2, 16) and (x₁, y₁) = (0, 0)
Soccer club makes the most money per car Compare the money earned for washing one car $5 < $8

Texas Go Math Grade 8 Lesson 6.3 H.O.T. Focus On Higher Order Thinking Answer Key

Question 15.
Draw Conclusions Gym A charges $60 a month plus $5 per visit. The monthly cost at Gym B is represented by y = 5x + 40, where x is the number of visits per month. What conclusion can you draw about the monthly costs of gyms?
Answer:
Since the rate per visit is the same, the monthly cost of Gym A is always more than Gym B

Comparing Functions Worksheet 8th Grade Answers Question 16.
Justify Reasoning Why will the value of y for the function y = 5x + 1 always be greater than that for the function y = 4x + 2 when x > 1?
Answer:
y₁ = 5x + 1 and y₂ = 4x + 2 Subtracting y₂ from y₁ we get
y₁ – y₂ = 5x + 1 – (4x + 2)
y₁ – y₂ = x – 1
For x ≥ 1 we get x – 1 ≥ 0
So, y₁ – y₂ ≥ 0
or y₁ ≥ y₂
Hence proved.

Question 17.
Analyze Relationships The equations of two functions are y = -21x + 9 and y = -24x + 8. Which function is changing more quickly? Explain.
Answer:
To determine among the given equations of functions changing more quickly, we have to determine the absolute
value of slope of the equations and then compare them. If the absolute value of the slope of one equation is greater than the slope of the other, then this equation changing more quickly than the other.

Determining the slopes of the two equations below.
y = -21x + 9
y = -24x + 8
Since, the two equations are already in the form of y = mx + b, where m is the slope, it is easier to determine their slope and these are
| Equation | Slope |
|–|–|
|y = -21x + 9 | -21|
|y = -24x + 81 -24|
Now, Let us get the absolute values of the two slopes.
|- 21| = 21
|-24| = 24

The absolute value of the slope of the equation y = -21x + 9 is 21 while of the equation y = -24x + 8 is 24. Since 24 is greater than 21, then the equation y = -24x + 8 changes more quickly than the equation y = -21x + 9.
See the explanation

Refer to our Texas Go Math Grade 8 Answer Key Pdf to score good marks in the exams. Test yourself by practicing the problems from Texas Go Math Grade 8 Module 4 Quiz Answer Key.

Texas Go Math Grade 8 Module 4 Quiz Answer Key

Texas Go Math Grade 8 Module 4 Ready to Go On? Answer Key

4.1 Representing Linear Nonproportional Relationships

Question 1.
Complete the table using the equation y = 3x + 2.

Answer:
In the given equation y = 3x + 2 include the given values of x.
x₁ = -1 → y = 3 • (-1) + 2 = -3 + 2 → y = -1
x₂ = 0 → y = 3 • 0 + 2 = 0 + 2 → y₁ = 2
x₃ = 1 → y = 3 • 1 + 2 = 3 + 2 → y₁ = 5
x₄ = 2 → y = 3 • 2 + 2 = 6 + 2 → y₁ = 8
x₅ = 3 → y = 3 • 3 + 2 = 9 + 2 → y₁ = 11
Obtained values of y put in the tabla

4.2 Determining Slope and Y-intercept

Grade 8 Module 4 End of Module Assessment Answer Key Question 2.
Find the slope and y-intercept of the line in the graph.

Answer:
The following equation for finding the slope and y-intercept is y = mx + b
The slope presents the ‘m’ in the equation, and the y-intercept represents b
The slope is calculated by:
m = \(\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\)
From the given graph, find two points that will help you calculate the slope.
I’ll take (0, 1) for (x₁, y₁) and (1, 4) for (x₂, y₂).
Thus: m = \(\frac{4-1}{1-0}\) = \(\frac{43}{1}\) = 3
To find the y-intercept, we need to include values of x and y from one of these two points and include the obtained slope. So:
y = mx + b
1 = 3 . 0 + b
1 = 0 + b
b = 1
m = 3, b = 1

4.3 Graphing Linear Nonproportional Relationships

Question 3.
Graph the equation y = 2x – 3 using slope and y-intercept.

Answer:
y = 2x – 3
Slope = 2
y-intercept = -3

Plot the point that contains the y-intercept:
(0, -3)
The slope is m = \(\frac{2}{1}\)
Use the slope to find a second point.
From (0, -3) count $2$ unit up and $1$ unit right. The new point is (1, -1)
Draw a line through the points

4.4 Proportional and Nonproportional Situations

Question 4.
Does the table represent a proportional or a nonproportional linear relationship?

Answer:
To see if some linear relationship is proportional or nonproportional, we have to use ratio \(\frac{y}{x}\), which needs to be constant for proportionality.

From the given table, we have values for x and y, so just put them into the calculation.
\(\frac{y}{x}\) = \(\frac{4}{1}\) = 4
= \(\frac{8}{2}\) = 4
= \(\frac{12}{3}\) = 4
= \(\frac{16}{4}\) = 4
= \(\frac{20}{5}\) = 4
Thus, the obtained values are constant so this table represents a proportional relationship.

4.5 Solving Systems of Linear Equations by Graphing

Question 5.
A school band ordered hats for $3 and large T-shirts for $5. They bought 150 items in all for $590. Graph a system of equations to find how many hats and T-shirts the band ordered.
____________
Answer:
The graph represents the nonproportional relationship because the obtained b, i.e. y-intercept, is bigger than 0 (b /
= 0) and the line does not pass the origin. When the b = 0 and the line passes the origin, the linear relationship is proportional.

Essential Question

Grade 8 Math Module 4 Answer Key Question 6.
How can you identify a linear nonproportional relationship from a table, a graph, and an equation?
Answer:
The graph represents the nonproportional relationship because the b, i.e. y-intercept from the given equation, is
bigger than 0 (b ≠ 0) and the line does not pass the origin. When the b = 0 and the line passes the origin, the linear relationship is proportional.

Texas Go Math Grade 8 Module 4 Mixed Review Texas Test Prep Answer Key

Selected Response

Question 1.
The table below represents which equation?

(A) y = -x – 10
(B) y = -6x
(C) y = 4x – 6
(D) y = -4x + 2
Answer:
(C) y = 4x – 6

Explanation:
The table is represented by Option C From the table, you can see that the y-intercept (when x = 0) is b = -6. Comparable to y = mx + b
y = 4x – 6

Question 2.
The graph of which equation is shown below?

(A) y = -2x + 3
(B) y = -2x + 1.5
(C) y = 2x + 3
(D) y = 2x + 1.5
Answer:
(A) y = -2x + 3

Explanation:
Option C is rejected
Since the graph is slanting downwards, the slope is negative.
The graph represents
y = -2x + 3

Texas Go Math 8th Grade Module 4 Answer Key Pdf Question 3.
The table below represents a linear relationship.

What is the y-intercept?
(A) -4
(B) -2
(C) 2
(D) 3
Answer:

The y-intercept is -2.
Option B is the correct answer.

Question 4.
Which equation represents a nonproportional relationship?
(A) y = 3x + 0
(B) y = -3x
(C) y = 3x + 5
(D) y = \(\frac{1}{3}\)x
Answer:
(C) y = 3x + 5

Explanation:
Option C represents a non-proportional relationship
y = 3x + 5

For a non-proportional relationship, the equation is y = mx + b and b ≠ 0.

Go Math 8th Grade Pdf Module 4 Quiz Answer Key Question 5.
Which statement describes the solution of a system of linear equations for two lines with the same slope and the same y-intercept?
(A) one nonzero solution
(B) infinitely many solutions
(C) no solution
(D) solution of 0
Answer:
(B) infinitely many solutions

Gridded Response

Question 6.
The table shows a proportional relationship. What is the missing y-value?

Answer:

Refer to our Texas Go Math Grade 8 Answer Key Pdf to score good marks in the exams. Test yourself by practicing the problems from Texas Go Math Grade 8 Lesson 5.3 Answer Key Linear Relationships and Bivariate Data.

Texas Go Math Grade 8 Lesson 5.3 Answer Key Linear Relationships and Bivariate Data

Essential Question
How can you contrast linear and nonlinear sets of bivariate data?

Your Turn

Find the equation of each linear relationship.

Question 1.

Answer:
Slope = \(\frac{60-40}{10-5}\) = \(\frac{20}{5}\) = 4
Find the slope using two points from the graph by Slope(m) = \(\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\) where (x₂, y₂) = (10, 60) and (x₁, y₁) = (5, 40)
b = 20 Read the y-intercept from the graph (when x = 0)
y = 4x + 20 Substituting the value of slope (m) and y-intercept in slope-intercept form.
y = mx + b
y = 4x + 20

Lesson 5.3 Answer Key Go Math 8th Grade Pdf Question 2.

Answer:
Slope = \(\frac{3600-480}{15-2}\) = \(\frac{3120}{13}\) = 240
Find the slope using two points from the graph by Slope(m) = \(\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\) where (x₂, y₂) = (15, 3600) and (x₁, y₁) = (2, 480)
b = 20 Read the y-intercept from the graph (when x = 0)

y = 240x Substituting the value of slope (m) and y-intercept in slope-intercept form. y = mx + b
y = 240x

Reflect

Question 3.
What If? Suppose a regulation changes the cost of the taxi ride to $1.80 per mile, plus a fee of $4.30. How does the price of the 6.5-mile ride compare to the original price?
Answer:
y = 1.8x + 4.3 Substituting the value of slope (m) and y-intercept in slope-intercept form.
y = mx + b
where m = 1.8 and b = 4.3

y = 1.8(6.5) + 4.3 Substitute x = 6.5 mile
y = 11.7 + 4.3 = 16
The cost remains the same.

Lesson 5.3 Answer Key 8th Grade Question 4.
How can you use a graph of a linear relationship to predict a value for a new input?
Answer:
Use the two points from the graph to determine the slope. Determine the y-intercept from the graph Substitute the m and b in slope-intercept form: y = mx + b
Substitute x in the equation to find y.

Question 5.
How can you use a table of linear data to predict a value?
Answer:
Use the two points from the table to determine the slope. Determine the y-intercept from slope and given point by substituting in slope-intercept form: y = mx + b
Write the equation. Substitute x in the equation to find y.

Your Turn

Paulina’s income from a job that pays her a fixed amount per hour is shown in the graph. Use the graph to find the predicted value.

Question 6.
Income earned for working 2 hours ______
Answer:
From the graph, we can read that in 2 hours, her income is 30.Butlet’strytogetitbycalculatingtheslopeand y$-intercept

From the graph, we can see that the line passes through the origin, which means that this linear relationship is proportional.
Therefore our starting point is (0, 0), and what else we can conclude is that our y-intercept (b) in this case is 0. → b = 0
The slope is calculated by:
m = \(\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\)
Choose two points from the graph and put them into the calculation. I’ll take (4, 60) for (x₁, y₁) and (6, 90) for (x₂, y₂). Thus
m = \(\frac{90-60}{6-4}\)
m = \(\frac{30}{2}\)
m = 15

We need to find the value of y, income earned for working 2 hours. x represents the time in hours. The standard form of the linear equation is y = mx + b, so include the obtained values of x, the slope, and the y-intercept.
y = 15 . 2 +0
y = 30 + 0
y = 30
To conclude, for 2 hours of work Paulina gets $30

Go Math Grade 5 Lesson 5.3 Answer Key Question 7.
Income earned for working 3.25 hours ______
Answer:
From the graph, we can see that the line passes through the origin, which means that this linear relationship is proportional.
Therefore our starting point is (0, 0), and what else we can conclude is that our y-intercept (b) in this case is 0. → b = 0
The slope is calculated by:
m = \(\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\)
Choose two points from the graph and put them into the calculation. I’ll take (4, 60) for (x₁, y₁) and (6, 90) for (x₂, y₂). Thus
m = \(\frac{90-60}{6-4}\)
m = \(\frac{30}{2}\)
m = 15

We need to find the value of y, income earned for working 3.5 hours. x represents the time in hours. The standard form of the linear equation is y = mx + b, so include the obtained values of x, the slope, and the y-intercept.
y = 15 . 3.5 + 0
y = 52.5 + 0
y = 52.5
$
To conclude, for 3.5 hours of work Paulina gets $52.5

Question 8.
Total income earned for working for five 8-hour days all at the standard rate ______
Answer:
Income after 8 hr = $120 Estimated from the graph, when x = 8hrs
Income after 5 – 8hrs = 120 * 5 = $600

Texas Go Math Grade 8 Lesson 5.3 Explore Activity Answer Key 

Contrasting Linear and Nonlinear Data
Bivariate data is a set of data that is made up of two paired variables. If the relationship between the variables is linear, then the rate of change (slope) is constant. If the graph shows a nonlinear relationship, then the rate of change varies between pairs of points.

Andrew has two options in which to invest $200. Option A earns a simple interest of 5%, while Option B earns an interest of 5% compounded annually. The table shows the amount of the investment for both options over 20 years. Graph the data and describe the differences between the two graphs.

Step 1
Graph the data from the table for Options A and B on the same coordinate grid.

Step 2
Find the rate of change between pairs of points for Option A and classify the relationship.

The rate of change between the data values is ____, so the graph of Option A shows a __________ relationship.

Step 3
Find the rate of change between pairs of points for Option B and classify the relationship.

The rate of change between the data values is ____,
so the graph of Option B shows a ____ relationship.

Reflect

Question 9.
Why are the graphs drawn as lines or curves and not discrete points?
Answer:
The number of years can be in fractions, hence graphs are drawn as lines or curves and not discrete points

Lesson 5.3 Answer Key 8th Grade Linear Relationships and Bivariate Data Question 10.
Can you determine by viewing the graph if the data have a linear or nonlinear relationship? Explain.
Answer:
Yes. Linear relationship is represented by a straight line and non-linear relationship is represented by a curve.

Question 11.
Draw Conclusions Find the differences in the account balances to the nearest dollar at 5-year intervals for Option B. Flow does the length of time that money is in an account affect the advantage that compound interest has over simple interest?
Answer:
Difference (0 – 5) = 255.26 – 200 = 55.26
Difference (5 – 10) = 325.78 – 255.26 = 70.52
Difference (10 – 15) = 415.79 – 325.78 = 90.01
Difference (15 – 20) = 530.66 – 415.79 = 114.87
The amount of interest earned increases with the number of years for compound interest.

Texas Go Math Grade 8 Lesson 5.3 Guided Practice Answer Key 

Use the following graphs to find the equation of the linear relationship. (Example 1)

Question 1.

Answer:

Use the given points to show that it is a linear relationship
Slope = \(\frac{60-30}{2-1}\) = \(\frac{30}{1}\) = 30 Find the slope using two points from the graph by Slope(m) = \(\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\) where (x₂, y₂) = (2, 60) and (x₁, y₁) = (1, 30)
b = 0 Read the y-intercept from the graph (when x = 0)
y = 30x Substituting the value of slope (m) and y-intercept in slope-intercept form.
y = mx + b
y = 30x

Question 2.

Answer:

Use the given points to show that it is a linear relationship.
Slope = \(\frac{12-7}{4-2}\) = \(\frac{5}{2}\) = 2.5 Find the slope using two points from the graph by Slope(m) = \(\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\) where (x₂, y₂) = (4, 12) and (x₁, y₁) = (2, 7)
b = 2 Read the y-intercept from the graph (when x = 0)
y = 2.5x + 2 Substituting the value of slope (m) and y-intercept in slope-intercept form. y = mx + b
y = 2.5x + 2

Linear Relationships and Bivariate Data Answer Key Question 3.
The graph shows the relationship between the number of hours a kayak is rented and the total cost of the rental. Write an equation of the relationship. Then use the equation to predict the cost of a rental that lasts 5.5 hours. (Example 2)


Answer:
Write the equation of the linear relationship.
(2, 70) and (4, 110)(Select two points)
m = \(\frac{110-70}{4-2}\) (Calculate the rate of change)
= \(\frac{40}{20}\) (Simplify)
= 20
y = mx + b
70 = 20(2) + b (Fill in values for x, y and m)
70 = 40 + b (Simplify)
30 = b (Solve for b)
The equation of the linear relationship is y = 20x + 30.

Use your equation from Step 1 to predict the cost of rental that last 5.5 hours.
y = 20x + 30 (Substitute x=5.5)
y = 20(5.5) + 30
y = 110 + 30 (Solve for y)
y = 140
The cost of a rental that lasts 5.5 hours is \140$.

Does each of the following tables represent a linear relationship? Why or why not? (Explore Activity)

Question 4.

Answer:
Yes The graph has a constant rate of change

8th Grade Go Math Lesson 5.3 Answer Key Question 5.

Answer:
No
The graph does not have a constant rate of change

Essential Question Chk-In

Question 6.
How can you tell if a set of bivariate data shows a linear relationship?
Answer:
It is a linear relationship if the rate of change is constant or the graph is a straight line

Texas Go Math Grade 8 Lesson 5.3 Independent Practice Answer Key 

Does each of the following tables represent a linear relationship? Why or why not?

Question 7.

Answer:
Rate of change = \(\frac{45-15}{9-3}\) = \(\frac{30}{6}\) = 5 Find the rate of change using
Rate of change = \(\frac{105-45}{21-9}\) = \(\frac{60}{12}\) = 5
Slope (m) = \(\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\)
It is a linear relationship as the rate of change is constant.

Question 8.

Answer:
Rate of change = \(\frac{76.8-30}{8-5}\) = \(\frac{46.8}{3}\) = 15.6 Find the rate of change using Slope (m) = \(\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\)
Rate of change = \(\frac{235.2-76.8}{14-8}\) = \(\frac{158.4}{6}\) = 26.4
It is a linear relationship as the rate of change is constant.

Explain whether or not you think each relationship is linear.

Question 9.
the cost of equal-priced DVDs and the number purchased
Answer:
Explanation A:
the relation between the cost of equal-priced DVDs and the number purchased is linear because the change rate
is constant. If we denote with e the cost of a DVD, with x the number of purchased DVDs, and with y the total cost of the purchased DVDs, we get an equation of the form:
y = cx

Explanation B:
Linear relationship The cost of DVDs increases at a constant rate with the number of purchases.

Go Math Grade 8 Lesson 5.3 Answer Key Question 10.
the height of a person and the person’s age
Answer:
Non – Linear relationship The height of a person does not increase at a constant rate with a person’s age

Question 11.
the area of a square quilt and its side length
Answer:
We can say that a relationship is linear if the rate of change or the slope is constant

In the relationship between the area of a square quilt and its side length, the independent variable is the side length and the dependent variable is the area. Now, say, for example, the side of the quilt is 4 feet 3 feet 2 feet its length will be 42, 32, and 22 square feet, respectively.

Now, we have these set of ordered pairs: {(2, 4), (3, 9), (4, 16)}. Based on this example, we have the following as possible solutions for the slope:
when we use {(2, 4), (3, 9)}, slope is
\(\frac{9-4}{3-2}[latex] = 5
when we use {(3, 9), (4, 16)}, slope is
[latex]\frac{16-9}{4-3}[latex] = 7
Looking at the slope, they are not constant, as the area and the sides increase, the rate of change also increases, thus, this is a nonlinear relationship

See the explanation.

Question 12.
the number of miles to the next service station and the number of kilometers
Answer:
We can say that a relationship is linear if the rate of change or the slope is constant.

In the relationship of the number of miles to the next service station and the number of kilometers, the independent variable is a number of kilometers and the dependent variable is the number of miles.

Now, for these values of the number of kilometers, {1, 2, 3, 4}, the corresponding number of miles are:
{0.621, 1.21, 1.86, 2.485}
when we use {(1, 0.61), (2, 1.243)}, slope is
[latex]\frac{1.24-0.62}{2-1}\) = 0.62
when we use {(3, 1.86), (4, 2.48)}, slope is
\(\frac{2.48-1.86}{4-3}\) = 0.62
Looking at the slope, they are constant, as the area and the sides increase, the rate of change remains the same, thus, this is a linear relationship

See the explanation.

Lesson 5.3 Go Math 8th Grade Answer Key Question 13.
Multistep The Mars Rover travels 0.75 feet in 6 seconds. Add the point to the graph. Then determine whether the relationship between distance and time is linear, and if so, predict the distance that the Mars Rover would travel in 1 minute.
Answer:
The black points on the chart are the default. The green point is what we were supposed to plot

To see if our linear relationship ¡s proportional, calculate the rate of change from the given information. To know if some linear relationship is proportional or nonproportional, we have to use ratio m = \(\frac{y}{x}\), which needs to be constant for proportionality.

We have 3 points on the graph: (4, 0.5); (6, 0.75) and (12, 1.5). Put those in the calculation:
m = \(\frac{0.5}{4}\) = 0.125
m = \(\frac{0.75}{6}\) = 0.125
m = \(\frac{1.5}{12}\) = 0.125

The rate of change is constant so this relationship is proportional. If the relationship is proportional we can
immediately conclude that our y-intercept is equal to 0 (b = 0).

The standard form of the linear equation is y = mx + b.
Our equation in slope-intercept form is:
y = 0.125x + 0 → y = 0.125x
y represents the distance (ft), and x represents the time (s).

We need to determine the distance that the Mars Rover would travel in 1 minute. Since the time is measured in seconds, we need to convert the minute to the default unit So 1 minute has 60 seconds. Put it in the equation to
get the distance.
y = 0.125 • 60
y = 7.5
In one minute, the Mars Rover would travel 7.5 ft.

Question 14.
Make a Conjecture Zefram analyzed a linear relationship, found that the slope-intercept equation was y = 3.5x + 16, and made a prediction for the value of y for a given value of x. He realized that he made an error calculating the the y-intercept and that it was actually 12. Can he just subtract 4 from his prediction if he knows that the slope is correct? Explain.
Answer:
Zefram used y = 3.5x + 16 for the prediction of y for a given value of æ. Eventually, he found out that he made an error calculating the y-intercept and that it was actually 12

Now, if he wants to make a correction to his prediction knowing that the slope is correct, he can simply subtract the predicted value by 4. Look at these illustrations.
y = 3.5x + 16 → y = (3.5x – 16) 4 → y = 3.5x + 12

Since subtracting 4 from the predicted value of y would make the y-intercept equal to 12(as shown in the illustrations), then Zefram can subtract 4 from the predicted value of y. So the answer is Yes.

See the explanation.

Texas Go Math Grade 8 Lesson 5.3 H.O.T. Focus On Higher Order Thinking Answer Key 

Question 15.
Communicate Mathematical Ideas The table shows a linear relationship. How can you predict the value of y when x = 6 without finding the equation of the relationship?

Answer:
The only calculation in which we have values for both x and y is the slope calculation.
Since y is unknown and we have to find its value, first we must calculate the slope to include its value as well in the
calculation. Otherwise, both the slope and y would be unknown.

So, to get the slope, choose two points from the table:
m = \(\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\)
m = \(\frac{114-76}{12-8}\)
m = \(\frac{38}{4}\)
m = 9.5

To get y, when X = 6, choose one point from the table, include it into the calculation, include value of x and the slope.

The obtained value of y is 57 when x = 6. → (6, 57).

Texas Go Math Grade 8 Lesson 5.3 Linear Relationships and Bivariate Data Question 16.
Critique Reasoning Louis says that if the differences between the values of x are constant between all the points on a graph, then the relationship is linear. Do you agree? Explain.
Answer:
If the relationship between the variables is linear, then the rate of change(slope) is constant The slope is simply the difference between the y values divided by the difference of the corresponding x values.

Louis says that if the differences between the values of x are constant between all the points on a graph, then the relationship is linear. I do not agree with their idea of Louis since some of the values of x may not be the same difference as the other points. For example, x values of a Linear equation are {1, 3, 7, 8}, the difference between 1 and 3 is different from the difference between 7 and 8, and so on.

See the explanation.

Question 17.
Make a Conjecture Suppose you know the slope of a linear relationship and one of the points that its graph passes through. How could you predict another point that falls on the graph of the line?
Answer:
Find the equation of the Linear relationship using the slope and given point. Then insert any x-value to find a y-value on the graph of the line.

Go Math Grade 8 Pdf Linear Relationships Unit Test Answer Key Question 18.
Explain the Errors Thomas used (7, 17.5) and (18, 45) from a graph to find the equation of a linear relationship as shown. What was his mistake?
m = \(\frac{45-7}{18-17.5}\) = \(\frac{38}{0.5}\) = 79
y = 79x + b
45 = 79 • 18 + b
45 = 1422 + b, so b = -1377
The equation is y = 79x – 1377.
Answer:
Thomas used (7, 17.5) and (18, 45) from the graph to find the linear relationship. Let us examine his mistake in his solution below.
m = \(\frac{45-7}{18-17.5}\) = \(\frac{38}{0.5}\) = 79
y = 79x + b
45 = 79 • 18 + b
45 = 1422 + b, so b = -1377
The equation is y = 79x – 1377

As we can notice, there is a mistake on the substitution of values of the variables in getting the slope m. The formula for finding the slope is
m = \(\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\)
In the problem, the points are (7, 17.5) and (18, 45), wherein
x₁ = 7, y₁ = 17.5, x₂ = 18, y₂ = 45
Instead of substituting 17.5 as the value of y, Thomas substituted 7 and instead of substituting 7 as the value of x₁, he substituted 17.5. In short, Thomas interchanged the values of x₁ and y₁.

Since Thomas made a mistake in the first step of the solution, the proceeding solutions are also incorrect.

Here is the correct answer for the given problem in the statement.
Substituting the values of the variables gives the slope of

Solving for the y-intercept, where in = 2.5 using the point (18, 45)
y = 2.5x + b
45 = 2.5(18)+b
45 = 45 + b
b = 45 – 45
b = 0

Now, having the slope m = 2.5 and y-intercept b = 0, the equation is
y = 2.5x + 0
y = 2.5x

Refer to our Texas Go Math Grade 8 Answer Key Pdf to score good marks in the exams. Test yourself by practicing the problems from Texas Go Math Grade 8 Lesson 4.2 Answer Key Determining Slope and y-Intercept.

Texas Go Math Grade 8 Lesson 4.2 Answer Key Determining Slope and y-Intercept

Essential Question
How can you determine the slope and the y-intercept of a line?

Texas Go Math Grade 8 Lesson 4.2 Explore Activity Answer Key 

Investigating Slope and y-intercept
The graph of every nonvertical line crosses the y-axis. The y-intercept is the y-coordinate of the point where the graph intersects the y-axis. The x-coordinate of this point is always 0.

The graph represents the linear equation y = –\(\frac{2}{3}\) + 4.

Step 1
Find the slope of the line using the points (0, 4) and (-3, 6)

Step 2
The line also contains the point (6, 0). What is the slope using (0, 4) and (6, 0)? Using (-3, 6) and (6, 0). What do you notice?

Step 3
Compare your answers in Steps 1 and 2 with the equation of the graphed line.

Step 4
Find the value of y when x = 0 using the equation y = –\(\frac{2}{3}\)x + 4. Describe the point on the graph that corresponds to this solution.

Step 5
Compare your answer in Step 3 with the equation of the line.

Your Turn

Find the slope and y-intercept of the line represented by each table.

Question 1.

Answer:
\(\frac{32-22}{4-2}\) = 5 Finding the slope using given points by
\(\frac{42-32}{6-4}\) = 5 slope (m) = (y₂ – y₁) ÷ (x₂ – x₁)
\(\frac{52-42}{8-6}\) = 5
Slope(m) = $5$
Work backward from x = 2 to x = 0 Find the initial value when the value of x is 0
\(\frac{32-22}{4-2}\) = \(\frac{10}{2}\)
x = 2 – 2 = 0 Subtract the difference of x and y from the first point
y = 22 – 10 = 12
y-intercept (b) 12

Lesson 4.2 Answer Key Go Math Grade 8 Pdf Question 2.

Answer:
The slope is calculated by:
m = \(\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\)
Thus:

So m = 7
To get the y-intercept (“b”) of the line, we need to include values of x and y in the equation:
y = mx + b
Include values of x and y from one point, I will take the first one.
8 = 7 . 1 + b
b = 8 – 7
b = 1

Explore Activity 2

Deriving the Slope-intercept Form of an Equation

In the following Explore Activity, you will derive the slope-intercept form of an equation.

Step 1
Let L be a line with slope m and y-intercept b. Circle the point that must be on the line. Justify your choice.
(b, 0) (0, b) (0, m) (m, 0)

Step 2
Recall that slope is the ratio of change in y to change in x. Complete the equation for the slope m of the line using the y-intercept (0, b) and another point (x, y) on the line.

Step 3
In an equation of a line, we often want y by itself on one side of the equation. Solve the equation from Step 2 for y.

Reflect

Question 3.
Critical Thinking Write the equation of a line with slope m that passes through the origin. Explain your reasoning.
Answer:
y= x When the line passes through the origin, y-intercept = 0 in
y = mx + b

Texas Go Math Grade 8 Lesson 4.2 Guided Practice Answer Key 

Find the slope and y-intercept of the line in each graph. (Explore Activity 1)

Question 1.

Slope m = ___ y-intercept b = ___
Answer:
The slope is calculated by:
m = \(\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\)
We have point one (0, 1) and point two (2, -3). Include them in the calculation:
m = \(\frac{-3-1}{2-0}\) = \(\frac{-4}{2}\) = -2
m = -2

According to the equation y = mx + b we can calculate the y-intercept, that is b. Include the values of x and y from one point and the obtained slope value
y = mx + b
-1 = -2 . 0 + b
b = 1 – 0
b = 1

Determining Slope and Y-Intercept Lesson 4.2 Answer Key Question 2.

Slope m = ___ y-intercept b = ___
Answer:
The scope is calculated by:
m = \(\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\)
We have point one (0,-15) and point two (3,0). Include them into the calculation:
m = \(\frac{0-(-15)}{3-0}\) = \(\frac{0+15}{3}\) = \(\frac{15}{3}\) = 5
m = 5
According to the equation y = mx + b we can calculate the y-intercept, that is b. Include the values of x and y from one point and the obtained slope value.
y = mx + b
-15 = 5 . 0 + b
b = -15

Question 3.

Slope m = ___ y-intercept b = ___
Answer:
The slope is calculated by:
m = \(\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\)
First, read the points on the graph, the places where the graph intersects a certain values of the x and y axes. The first one is (0, -2) and the second one is (2, 1). Include them into the calculation:
m = \(\frac{1-(-2)}{2-0}\) = \(\frac{1+2}{2}\) = \(\frac{3}{2}\)
m = \(\frac{3}{2}\)

According to the equation y = mx + b we can calculate the y-intercept, that is b. Include the values of x and y from one point and the obtained slope value.
y = mx + b
-2 = \(\frac{3}{2}\) . 0 + b
-2 = 0 + b
b = -2

Lesson 4.2 Determining Slope and Y-Intercept Answer Key Pdf Question 4.

Slope m = ___ y-intercept b = ___
Answer:
The slope is calculated by:
m = \(\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\)
First, read the points on the graph, the places where the graph intersects a certain value of the x and y axes
The first one is (0,9) and the second one is (3,0). Include them into the calculation:
m = \(\frac{0-9}{3-0}\) = \(\frac{-9}{3}\) = -3
m = -3

According to the equation y = mx + b we can calculate the y-intercept, that is b. Include the values of x and y from one point and the obtained slope value.
y = mx + b
9 = -3 . 0 + b
9 = 0 + b
b = 9

Find the slope and y-intercept of the line represented by each table. (Example 1)

Question 5.

slope m = ______y-intercept b = ______
Answer:
m = \(\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\)
Include values of x and y from the table, for each and next point:

To get the y-intercept (“b”) of the line, we need to include values of x and y in the equation:
y = mx + b
Include values of x and y from any point
1 = 3 . 0 + b
1 = 0 + b
b = 1

Question 6.

Answer:
The slope is calculated by:
m = \(\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\)
Include values of x and y from the table, for each and next point:

To get the y-intercept (“b”) of the line, we need to include values of x and y in the equation:
y = mx + b
Include values of x and y from any point
140 = -4 . 0 + b
140 = 0 + b
b = 140

Essential Question Check-In

Go Math Answer Key Grade 8 Lesson 4.2 Determining Slope and Y-Intercept Question 7.
How can you determine the slope and the y-intercept of a line from a graph?
Answer:
Choose any two points on the line from the graph and use it to find the slope.
Determine the point where the line crosses the y-axis to find the y-intercept.

Texas Go Math Grade 8 Lesson 4.2 Independent Practice Answer Key 

Question 8.
Some carpet cleaning costs are shown in the table. The relationship is linear. Find and interpret the rate of change and the initial value for this situation.

Answer:
It is already written in the task that the relationship is linear, so the slope (rate of change) can be found with only
two points.

The slope is calculated by:
m = \(\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\)

The first point is (1, 125) and the second point is (2, 175). Put those in the calculation:
m = \(\frac{175-125}{2-1}\) = \(\frac{50}{1}\) = 50
The standard form of our Linear function is
y = mx – 1- b
We need to get the initial value of this function.
Take a look at the equation, compare the data we already have, and find what we need to get. Thus, the y-intercept (”b”) is unknown.

Include the values of x, y, and the slope in the equation and calculate:
y = mx + b
125 = 50 . 1 + b
125 = 50 + b
b = 125 – 50
b = 75

Lesson 4.2 Slope of a Line Answer Key Go Math 8th Grade Pdf Question 9.
Make Predictions The total cost to pay for parking at a state park for the day and rent a paddleboat are shown.

a. Find the cost to park for a day and the hourly rate to rent a paddleboat.
Answer:
Slope = \(\frac{29-17}{2-1}\) = \(\frac{12}{1}\) = 12 Finding the slope using any two given points by Slope(m) = (y₂ – y₂) ÷ (x₂ – x₁)
where
(x₂, y₂) = (2, 29) and (x₁, y₁) = (1, 17)
The hourly rent is $\$12$ per hr

Work backward from x = 1 to x = 0 Find the initial value when the value of x is 0
\(\frac{29-17}{2-1}\) = \(\frac{12}{1}\)
x = 1 1 = 0 Subtract the difference of x and y from the first point.
y = 17 – 12 = 5
The cost to park for a day is $\$5$

b. What will Lin pay if she rents a paddleboat for 3.5 hours and splits the total cost with a friend? Explain.
Answer:
Total Cost = 3.5(12) + 5 = 47 When Lin paddles for 3.5 hr
Lin’s Cost = \(\frac{47}{2}\) = 23.5

Question 10.
Multi-Step Raymond’s parents will pay for him to take sailboard lessons during the summer. He can take half-hour group lessons or half-hour private lessons. The relationship between cost and number of lessons is linear.

a. Find the rate of change and the initial value for the group lessons.
Answer:
Slope = \(\frac{85-55}{2-1}\) = \(\frac{30}{1}\) = 30 Finding the slope using any two given points by
slope (m) = (y₂ – y₁) ÷ (x₂ – x₁)
Rate of change is $\$30$ per lesson
where (x₂, y₂) = (2, 85) and (x₁, y₁) = (1, 55)
Work backward from x = 1 to x = 0 Find the initial value when the value of x is 0
\(\frac{85-55}{2-1}\) = \(\frac{30}{1}\)
x = 1 – 1 = 0 Subtract the difference of x and y from the first point
y = 55 – 30 = 25
The initial value of group lessons is $\$25$

b. Find the rate of change and the initial value for the private lessons.
Answer:
Slope = \(\frac{125-75}{2-1}\) = \(\frac{50}{1}\) = 50 Finding the slope using any two given points by
slope (m) = (y₂ – y₁) ÷ (x₂ – x₁)
Rate of change is $\$50$ per lesson
where (x₂, y₂) = (2, 125) and (x₁, y₁) = (1, 75)
Work backward from x = 1 to x = 0 Find the initial value when the value of x is 0
\(\frac{125-75}{2-1}\) = \(\frac{50}{1}\)
x = 1 – 1 = 0 Subtract the difference of x and y from the first point
y = 75 – 50 = 25
The initial value of group lessons is $\$25$

c. Compare and contrast the rates of change and the initial values.
Answer:
The initial value for both types of lessons is the same. Comparing results from parts a and b.
The rate of change is higher for private lessons than group lessons.

Vocabulary Explain why each relationship is not linear.

Lesson 4.2 Determining Slope and Y-Intercept Reteach Answer Key Question 11.

Answer:
\(\frac{6.5-4.5}{2-1}\) Finding the rate of change using given points by
the slope (m) = (y₂ – y₁) ÷ (x₂ – x₁)
\(\frac{8.5-6.5}{3-2}\) = 2
\(\frac{11.5-8.5}{4-3}\) = 3
The rate of change is not constant, hence the relationship is not linear

Question 12.

Answer:
Calculate the slope for each and next point.

Question 13.
Communicate Mathematical Ideas Describe the procedure you performed to derive the slope-intercept form of a linear equation. (Explore Activity 2)
Answer:
Express the slope m between a random point (x, y) on the line and the point (0, b) where the line crosses the y-axis. Then solve the equation for y.

Texas Go Math Grade 8 Lesson 4.2 H.O.T. Focus On Higher Order Thinking Answer Key 

Question 14.
Critique Reasoning Your teacher asked your class to describe a real-world situation in which the y-intercept is 100 and the slope is 5. Your partner gave the following description: My younger brother originally had 100 small building blocks, but he has lost 5 of them every month since.

a. What mistake did your partner make?
Answer:
a. If the brother loses 5 blocks every month, the slope would be -5 and not 5.
When the initial value is decreasing, the slope is negative.

b. Describe a real-world situation that does match the situation.
Answer:
I bought a 1oo card pack and bought 5 additional cards every month. Real-world situation

8th Grade Go Math Answer Key Practice and Homework Lesson 4.2 Question 15.
Justify Reasoning John has a job parking cars. He earns a fixed weekly salary of $300 plus a fee of $5 for each car he parks. His potential earnings for a week are shown in the graph. At what point does John begin to earn more from fees than from his fixed salary? Justify your answer.

Answer:
After parking 60 cars, John’s earnings become $600 double his initial base salary of $300. Hence, after parking 61 cars, his earning from the fee becomes more than his fixed salary.
\(\frac{300}{5}\) = 60
He earns the same ¡n fees as his fixed salary for parking.

Refer to our Texas Go Math Grade 8 Answer Key Pdf to score good marks in the exams. Test yourself by practicing the problems from Texas Go Math Grade 8 Module 5 Answer Key Writing Linear Equations.

Texas Go Math Grade 8 Module 5 Answer Key Writing Linear Equations

Essential Question
How can you use linear equations to solve real world problems?

Texas Go Math Grade 8 Module 5 Are You Ready? Answer Key

Complete these exercises to review skills you will need for this chapter.

Write each fraction as a decimal.

Question 1.
\(\frac{1}{8}\) ________
Answer:
Given,
\(\frac{1}{8}\)
Convert from fraction to the decimal.
= 1 ÷ 8
= 0.125

Texas Go Math Grade 8 Answer Key Module 5 Answer Key Question 2.
\(\frac{0.3}{0.5}\) ________
Answer:
Given,
\(\frac{0.3}{0.5}\)
Convert from fraction to the decimal.
\(\frac{3}{10}\) ÷ \(\frac{5}{10}\) = \(\frac{3}{5}\) = 0.6

Question 3.
\(\frac{0.13}{0.8}\) ________
Answer:
Given,
\(\frac{0.13}{0.8}\)
Convert from fraction to decimal.
\(\frac{13}{10}\) ÷ \(\frac{8}{10}\) = \(\frac{13}{8}\) = 1.625

Question 4.
\(\frac{0.39}{0.52}\) ________
Answer:
Given,
\(\frac{0.39}{0.52}\)
Convert from fraction to the decimal.
\(\frac{39}{100}\) ÷ \(\frac{52}{100}\) = \(\frac{39}{52}\) = 0.75

Solve each equation using the inverse operation.

Question 5.
7p = 28 __________________
Answer:
7p = 28 p is multiplied by 7
\(\frac{7 p}{7}\) = \(\frac{28}{7}\) To solve the equation, use the inverse operation, division.
p = 4

Question 6.
h – 13 = 5 ________
Answer:
h – 3 = 13 is subtracted from h
h – 13 + 13 = 5 + 13 To solve the equation, use the inverse operation, addition.
h = 18

Go Math Grade 8 Module 5 Answer Key Question 7.
\(\frac{y}{3}\) = -6 ______
Answer:
\(\frac{y}{3}\) = -6 y is divided by 3
\(\frac{y}{3}\) * 3 = -6 * 3 To solve the equation, use the inverse operation, multiplication.
y = -18

Question 8.
b + 9 = 21 _________
Answer:
b + 9 = 21 9 is added to b
b + 9 – 9 = To solve the equation, use the inverse operation, subtraction.
b = 12

Question 9.
c – 8 = -8 ________________
Answer:
c – 8 = 8 8 is subtracted from c
c – 8 + 8 = -8 + 8 To solve the equation, use the inverse operation, addition
c = 0

Question 10.
3n = -12 _______
Answer:
3n = -12
n is multiplied by 3
\(\frac{3 n}{3}\) = \(\frac{-12}{3}\) To solve the equation, use the inverse operation, division.
n = -4

Question 11.
-16 = m + 7 _______
Answer:
-16 = m + 7 7 is added to m

-16 – 7 = m + 7 To solve the equation, use the inverse operation, subtraction.
m = -23

Grade 8 Math Module 5 Answer Key Question 12.
\(\frac{t}{-5}\) = -5 ____
Answer:
\(\frac{t}{-5}\) = -5
As we can see t is divided by (-5), so to solve the equation, we use the inverse operation: muLtiplication.
\(\frac{t}{-5}\) . (-5) = (-5) . (-5)
t = 25

Texas Go Math Grade 8 Module 5 Reading Start-Up Answer Key

Visualize Vocabulary

Use the ✓ words to complete the diagram. You can put more than one word in each bubble.

Understand Vocabulary

Complete the sentences using the preview words.

Question 1.
A set of data that is made up of two paired variables is ____
Answer:
A set of data that is made up of two paired variables is Bivariate data
Bivariate data is defined as a set of data that is made up of two o paired variables. Each value of one of the variables is paired with a value of the other variable.
By definition, a set of data that is made up of two variables is bivariate data.
Thus, the blank part of the statement is bivariate data.

Texas Go Math Grade 8 Answer Key Pdf Module 5 Question 2.
When the rate of change varies from point to point, the relationship is a ____.
Answer:
When the rate of change varies from point to point, the relationship is a nonlinear
A nonlinear relationship is a type of relationship between two points in which change in one point does not correspond with change in another point. It is shown when the rate of change varies between pairs of points.
By description, when the rate of change varies from point to point the relationship is nonlinear.
Thus, the blank part of the statement is nonlinear.

Refer to our Texas Go Math Grade 8 Answer Key Pdf to score good marks in the exams. Test yourself by practicing the problems from Texas Go Math Grade 8 Module 5 Quiz Answer Key.

Texas Go Math Grade 8 Module 5 Quiz Answer Key

Texas Go Math Grade 8 Module 5 Ready to Go On? Answer Key

5.1 Writing Linear Equations from Situations and Graphs

Write the equation of each line in slope-intercept form.

Question 1.

Answer:
The standard form of an equation for the linear function is y = mx + b. To write an equation in slope-intercept form, we need to get values for the slope and y-intercept.

We can read the starting point from the graph, which is the intersection on the y-axis when x = 0. So we can read what our value is for the y-intercept. → starting point is (0, b)
Thus, b = 20.

The slope is calculated by:
m = \(\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\)
Choose two points from the graph and put them in thecalculation I’ll take (0, 20) for (x₁, y₁) and (2, 80) for (x₂, y₂)
m = \(\frac{80-20}{2-0}\)
m = \(\frac{60}{2}\)
m = 30
The equation in slope-intercept form is: y = 30x + 20

Module 5 Test Answers Texas Go Math Grade 8 Pdf Question 2.

Answer:
The standard form of an equation for the linear function is y = mx + b. To write an equation in slope-intercept form, we need to get values for the slope and y-intercept.

We can read the starting point from the graph, which is the intersection on the y-axis when x = 0. So we can read what our value is for the y-intercept. → starting point is (0, b)
Thus, b = 60

The slope is calculated by:
\(\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\)

Choose two points from the graph and put them in the calculation (0, 60) for (x₁, y₁) and (6, 0) for (x₂, y₂)
m = \(\frac{0-60}{6-0}\)
m = \(\frac{-60}{6}\)
m = -10
The equation in slope-intercept form is: y = -10x + 60

5.2 Writing Linear Equations from a Table

Write the equation of each linear relationship in slope-intercept form.

Question 3.

Answer:
The standard form of an equation for the linear function is y = mx + b. To write an equation in slope-intercept form, we need to get values for the slope and y-intercept.

We can read the starting point from the graph, which is the intersection on the y-axis when x = 0. So we can read what our value is for the y-intercept. → starting point is (0, b)
Thus, b = 1.5

The slope is calculated by:
\(\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\)

Choose two points from the graph and put them in the calculation (0, 1.5) for (x₁, y₁) and (100, 36.5) for (x₂, y₂)
m = \(\frac{36.5-1.5}{100-0}\)
m = \(\frac{35}{100}\)
m = 0.35
The equation in slope-intercept form is: y = 0.35x + 1.5

Grade 8 Texas Go Math Module 5 Answer Key Question 4.

Answer:
The standard form of an equation for the linear function is y = mx + b. To write an equation in slope-intercept form, we need to get values for the slope and y-intercept.

The slope is calculated by:
\(\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\)

Choose two points from the graph and put them in the calculation (25, 94) for (x₁, y₁) and (35, 88) for (x₂, y₂)
m = \(\frac{88-94}{35-25}\)
m = \(\frac{-6}{10}\)
m = -0.6
To find the y-intercept (b), we can include one point from the table and the value of the slope in the standard form of the equation. So:
y = mx + b
94 = -0.6 . 25 + b
94 = -15 + b
b = 94 + 15
b = 109

The equation in slope-intercept form is: y = 0.6x + 109

5.3 Linear Relationships and Bivariate Data

Write the equation of the line that connects each set of data points.

Question 5.

Answer:
The standard form of an equation for the linear function is y = mx + b. To write an equation in slope-intercept form, we need to get values for the slope and y-intercept.

The slope is calculated by:
\(\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\)

Choose two points from the graph and put them in the calculation (20, 40) for (x₁, y₁) and (50, 60) for (x₂, y₂)
m = \(\frac{60-40}{50-20}\)
m = \(\frac{20}{30}\)
m = \(\frac{2}{3}\)
m ≈ 0.67

To find the y-intercept (b), we can include one point from the table and the value of the slope in the standard form of the equation. So:
y = mx + b
40 = 0.67 . 20 + b
40 = 13.4 + b
b = 40 – 13.4
b ≈ 26.6
The equation in slope-intercept form is: y = 0.67x + 26.6

Question 6.

Answer:

Confirm the points represent a linear relationship

Slope = \(\frac{50-65}{30-25}\) = \(\frac{-15}{5}\) = -3
Find the slope using given points from the table by Slope(m) = \(\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\) where
(x₂, y₂) = (30, 50) and (x₁, y₁) = (25, 65)
50 = -3(30) + b Substituting the value of slope (m) and (x, y) in the slope-intercept form to find y-intercept (b):
50 = -90 + b y = mx + b
b = 50 + 90 = 140
y = -3x + 140 Substitute the value of slope (m) and y-intercept (b) in slope-intercept form: y = mx + b

Essential Question

Go Math 8th Grade Pdf Module 5 Linear Functions Answer Key Question 7.
Write a real-world situation that can be represented by a linear relationship.
Answer:
A linear equation uses one or more variables where one variable is independent of the other. The degree of the linear equation is one.
Imagine that you are taking a cab while on vacation. You know that the cab service charges $10 to pick your family up from your hotel and another $0.20 per mile for the trip. Without knowing how many miles it will be to each destination, you can set up a linear equation that can be used to find the cost of any taxi trip you take on your trip. By using “x” to represent the number of miles to your destination and “y” to represent the cost of that taxi ride, the linear equation would be y = 0.20x + 10.

Texas Go Math Grade 8 Module 5 Mixed Review Texas Test Prep Answer Key

Selected Response

Question 1.
An hourglass is turned over with the top part filled with sand. After 3 minutes, there are 855 mm of sand in the top half. After 10 minutes, there are 750 mm of sand in the top half. Which equation represents this situation?
(A) y = 285x
(B) y = -10.5x + 900
(C) y = -15x + 900
(D) y = 75x
Answer:
(C) y = -15x + 900

Explanation:
(3, 855) Write the given information as ordered pair

(10, 750)

Slope = \(\frac{855-750}{3-10}\) = \(\frac{105}{-7}\) = -15 Find the slope using given points by Slope (m) = \(\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\)

750 = -15(10) + b Substituting the value of slope (m) and (x, y) in the
750 = -15(10) + b slope intercept form to find y-intercept (b):
750= -150 + b
b = 750 + 150 = 900y = mx + b
y = -15x + 900 Substitute the value of slope (m) and y-intercept (b) in slope-intercept form:
y = mx + b

Question 2.
Which graph shows a linear relationship?

Answer:

Graph B represents the linear relationship The data appears to lie on a straight line

8th Grade Math Module 5 Writing Linear Equations Module Quiz Answer Key Question 3.
What are the slope and y-intercept of the relationship shown in the table?

(A) slope = 0.05, y-intercept = 1,500
(B) slope = 0.5, y-intercept = 1,500
(C) slope = 0.05, y-intercept = 2,000
(D) slope = 0.5, y-intercept = 2,000
Answer:
(C) slope = 0.05, y-intercept = 2,000

Explanation:
slope = \(\frac{3000-2500}{20000-10000}\) = \(\frac{500}{10000}\) = 0.05
Find the slope using given points from the table by Slope(m) = \(\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\) where (x₂, y₂) = (20000, 3000) and (x₁, y₁) = (10000, 2500)

3000 = 0.05(2000) + b Substituting the value of slope (m) and (x, y) in the slope-intercept form to find y-intercept (b):
3000 = 1000 + b y = mx + b
b = 3000 – 1000 = 2000

Gridded Response

Question 4.
Franklin’s faucet was leaking water, so he put a bucket underneath to catch the water. After a while, Franklin started keeping track of how much water was leaking. His data is in the table below.

Predict how many gallons of water will have leaked if Franklin hasn’t stopped the leak after 14 hours.

10
Answer:
9.5 – 8 = 1.5
So add 1.5 and it will show you what you get in 14 hours.
In 14 hours if Franklin doesn’t stop the leak it will be 23 quarts

Refer to our Texas Go Math Grade 8 Answer Key Pdf to score good marks in the exams. Test yourself by practicing the problems from Texas Go Math Grade 8 Lesson 5.1 Answer Key Writing Linear Equations from Situations and Graphs.

Texas Go Math Grade 8 Lesson 5.1 Answer Key Writing Linear Equations from Situations and Graphs

Essential Question
How do you write an equation to model a linear relationship given a graph or a description?

Texas Go Math Grade 8 Lesson 5.1 Explore Activity Answer Key 

Explore Activity
Writing an Equation in Slope-intercept Form

Greta makes clay mugs and bowls as gifts at the Crafty Studio. She pays a membership fee of $15 a month and an equipment fee of $3.00 an hour to use the potter’s wheel, table, and kiln.

Write an equation in the form y = mx + b that Greta can use to calculate her monthly costs.

A. What is the input variable, x, for this situation?
What is the output variable, y for this situation?

B. During April, Greta does not use the equipment at all.
What will be her number of hours (x) for April? ______
What will be her cost (y) for April? ______
What will be the y-intercept, b, in the equation? _____

C. Greta spends 8 hours in May for a cost of $15 + 8($3) = ____
In June, she spends 11 hours for a cost of _________________
From May to June, the change in x-values is _______________
From May to June, the change in y-values is _______________
What will be the slope, m, in the equation? ______

D. Use the values for m and b to write an equation for Greta’s costs in the form y = mx + b: __________________

Reflect

Question 1.
What does the value of the slope represent in this context?
Answer:
The slope represents the rental for each DVD borrowed.

Go Math Grade 8 Lesson 5.1 Answer Key Question 2.
Describe the meaning of the y-intercept.
Answer:
The y-intercept represents the one-time membership fee.

Your Turn

Question 3.
The cash register subtracts $2.50 from a $25 Coffee Café gift card for every medium coffee the customer buys. Use the graph to write an equation in slope-intercept form to represent this situation.

Answer:
slope = \(\frac{0-25}{10-0}\) = \(\frac{-25}{10}\) = -2.5 Finding the slope using given points by Slope(m) = (y₂ – y₁) ÷ (x₂ – x₁) where (x₂, y₂) = (0, 25) and (x₁, y₁) = (10, 0)
y-intercept (b) = 25 From the graph
y = -2.5x + 25 Substituting the value of slope (m) and y-intercept in slope-intercept form:
y = mx + b
y = -2.5x + 25

Reflect

Question 4.
Without graphing, tell whether the graph of this equation rises or falls from left to right. What does the sign of the slope mean in this context?
Answer:
Whether the graph is ascending or descending depends on the sign of the slope. If the slope is positive then the graph is increasing, because we get the second point by counting upwards from the starting point (y-intercept, b) for the number written in the slope numerator. If the slope is negative then the graph is descending, because we get the second point by counting down from the starting point (y-intercept, b) for the number written in the slope numerator.

Your Turn

Lesson 5.1 Writing Linear Equations from Situations and Graphs Question 5.
Hari’s weekly allowance varies depending on the number of chores he does. He received $16 in allowance the week he did 12 chores, and $14 in allowance the week he did 8 chores. Write an equation for his allowance in slope-intercept form. _______
Answer:
Input variables: Number of chores
Output variables Weekly allowance Identify the input and output variables.
(12, 16) Write the information given in the problem as ordered pairs.
(8, 14)
Slope = \(\frac{16-14}{12-8}\) = \(\frac{2}{4}\) = \(\frac{1}{2}\) Finding the slope using given points by slope (m) = (y₂ – y₁) ÷ (x₂ – x₁)

14 = \(\frac{1}{2}\)(8) + b Substituting the value of slope (m) and (x, y) in the slope-intercept form to find y-intercept (b):
14 = 4 + b y = mx + b
b = 14 – 4 = 10
y = \(\frac{1}{2}\)x + 10 Substituting the value of slope (m) and y-intercept in slope-intercept form.

Texas Go Math Grade 8 Lesson 5.1 Guided Practice Answer Key 

Question 1.
Li is making beaded necklaces. For each necklace, she uses 27 spacers, plus 5 beads per inch of necklace length. Write an equation to find how many beads Li needs for each necklace. (Explore Activity)
a. input variable: _____________________________________________
Answer:
Input variable: The length of the necklace in inches.

b. output variable: _____________________________________
Answer:
Output variable: The total number of beads in the necklace.

c. equation: ______________
Answer:
Equation:
y = 5x + 27

Question 2.
Kate is planning a trip to the beach. She estimates her average speed to graph her expected progress on the trip. Write an equation in slope-intercept form that represents the situation. (Example 1)

Choose two points on the graph to find the slope.
m = \(\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\) = _______
Read the y-intercept from the graph: b = ______________
Use your slope and y-intercept values to write an equation in slope-intercept form. _________________________
Answer:
m = \(\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\) = \(\frac{0-300}{5-0}\) = \(\frac{-300}{5}\) = -60
Finding the slope using given points (x₂, y₂) = (0, 300) and (x₁, y₁) = (5, 0)
y-intercept (b) = 300 Read from the graph (when x = 0)
y = -60x + 300 Substituting the value of slope (m) and y-intercept in slope-intercept form:
y = mx + b

Texas Go Math Grade 8 Writing Linear Equations from Graphs 5.1 Answer Key Question 3.
At 59°F, crickets chirp at a rate of 76 times per minute, and at 65°F, they chirp 100 times per minute. Write an equation in slope-intercept form that represents the situation. (Example 2)
Input variable: _____________________
Output variable: __________________________
m = \(\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\) = __________
Substitute in y = mx + b: _______________ + b; __________ = b
Write an equation in slope-intercept form. ___________________
Answer:
To determine the equation in slope intercept form y = mx + b that represents the situation At 59° F crickets chirp at a rate of 76 times per minute, and at 65° F, they chirp 100 times per minute, let us follow the following steps.

First, let us identify the input and output variables. The input variable x is the temperature while the output variable y is the number of chirps per minute.

Second, let us write the ordered pairs.
at 59 °F, crickets chirp at a rate of 76 times per minute:(59, 76)
at 65 °F, crickets chirp at a rate of 100 times per minute:(65, 100)
x₁ = 59°
y₁ = 76
x₂ = 65°
y₂ = 100

Third, let us determine the slope in using the formula
m = \(\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\)
Substituting the values of x₁, y₁, x₂ and y₂.
m = \(\frac{100-76}{65-59}\)
m = \(\frac{24}{6}\)
m = 4

Fourth, let us find the value of the y-intercept b. Using the variables and the slope 4, we can have two equations
for us to find b, these are
76 = 4 • 59 + b
and
100 = 4 • 65 + b
We can use any of these two to determine the value of b, that is
100 = 4 • 65
100 = 260 + b
b = 100 – 260
b = -160

Now, let us form the equation in slope-intercept form, having the slope m = 4 and y-intercept b = -160, that is
y = mx + b
y = 4x – 160
see the explanation

Essential Question Check-In

Question 4.
Explain what m and b in the equation y = mx + b tell you about the graph of the line with that equation.
Answer:
m indicates the steepness of the line and whether the graph is rising or falling (left from right). b indicates the point where the line crosses the y-axis.

Texas Go Math Grade 8 Lesson 5.1 Independent Practice Answer Key 

Question 5.
A dragonfly can beat its wings 30 times per second. Write an equation in slope-intercept form that shows the relationship between flying time in seconds and the number of times the dragonfly beats its wings.
Answer:
Input variable: Flying time in seconds Identify the input and output variables.
Output variable: Number of times the dragonfly beats its wing
y = 30x Substituting the value of slope (m) and y-intercept in slope-intercept form.
where m = 30 times per second and b = 0

Go Math 8th Grade Pdf Lesson 5.1 Equations and Their Graphs Question 6.
A balloon is released from the top of a platform that is 50 meters tall. The balloon rises at the rate of 4 meters per second. Write an equation in slope-intercept form that tells the height of the balloon above the ground after a given number of seconds.
Answer:
The problem is asking for the equation that tells the height of the balloon above the ground after a given number of seconds in slope-intercept form.

The statement that the balloon is released from the top of a platform that is 50 meters tall and rises at the rate of 4 meters per second
Let
y be the height of the balloon above the ground, x be the number of seconds, and 4s be the rate of the balloon as it rises per second.

Using the representation and the given, we should equate the height of the balloon above the ground y to the rate of the balloon as it rises per second 4x added to 50 which is the height of the platform, which gives us the equation,
y = 4x + 50

The graph shows a scuba diver’s ascent over time.

Question 7.
Use the graph to find the slope of the line. Tell what the slope means in this context.
______________________
Answer:
First, we choose two points on the graph to find the slope. We substitute (0, -10) for (x₁, y₁) and (80, 0) for (x₂, y₂)

Therefore, the slope equals to 0.125 means that the scuba driver ascends 0.125 meters each second.

Question 8.
Identify the y-intercept. Tell what the y-intercept means in this context.
Answer:
The starting point on the graph is (0, -10), i.e. (0, b). So the value of the y-intercept is -10. Reading from the graph, the y-intercept is the value for the depth (in meters) from which the scuba diver started to swim to the surface.

Question 9.
Write an equation in slope-intercept form that represents the diver’s depth over time.
Answer:
From the given graph, find two points that will help you calculate the slope.
I’ll take (0, -10) for (x₁, y₁) and (80, 0) for (x₂, y₂).
Thus:

The starting point on the graph is (0, -10), ie. (0, b). So the value of the y-intercept is -10. → b = -10

The standard form of the linear equation is y = mx + b, so include the obtained values of the slope and y-intercept.
y = \(\frac{1}{8}\)x – 10

Texas Go Math Grade 8 Pdf Lesson 5.1 Answer Key Question 10.
The formula for converting Celsius temperatures to Fahrenheit temperatures is a linear equation. Water freezes at 0°C, or 32°F, and it boils at 100°C, or 212°F. Find the slope and y-intercept for a graph that gives degrees Celsius on the horizontal axis and degrees Fahrenheit on the vertical axis. Then write an equation in slope-intercept form that converts degrees Celsius into degrees Fahrenheit.
Answer:
To determine the equation in slope intercept form y = mx + b that converts degrees Celsius into degrees. Fahrenheit, let us follow the following steps by finding the slope and y-intercept form given that the water freezes at 0°C or 32°F and it boils at 100°C or 212°F

First, let us identify the input and output variables. The input variable x is the temperature in degrees Celsius while the output variable y is the temperature in degrees Fahrenheit

Second, let us write the ordered pairs.
water freezes at 0°C or 32°F:(0, 32)
water boils at 100°C or 212°F:(100, 212)
x₁ = 0
y₁ = 32
x₂ = 100
y₂ = 212

Third, let us determine the slope ni using the formula
m = \(\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\)
Substituting the values of x₁, y₁, x₂, and y₂
m = \(\frac{212-32}{100-0}\)
m = \(\frac{180}{100}\)
m = 1.8

Fourth, let us find the value of the y-intercept b. Using the variables and the slope 4, we can have two equations for us to find b, these are
32 = 1.8 . 0 + b
and
212 = 1.8 . 100 + b
We can use any of these two to determine the value of b, that is
212 = 1.8 . 100 + b
212 = 180 + b
b = 212 – 180
b = 32
Now, let us form the equation in slope-intercept form, having the slope m = 1.8 and y-intercept b = 32, that is
y = mx + b
y = 1.8x + 32

See the explanation.

Question 11.
The cost of renting a sailboat at a lake is $20 per hour plus $12 for lifejackets. Write an equation in slope-intercept form that can be used to calculate the total amount you would pay for using this sailboat.

Answer:
To determine the equation in slope-intercept form to calculate the total amount you would pay for using a sailboat given that the cost of renting a sailboat at a lake is $20 per hour plus $12 for life jackets, let us represent the different variables in the problem.

Let y be the total cost you would pay for using a sailboat, x be the number of hours you would use the sailboat, $20 which is the cost of renting a sailboat per hour be the slope, and $12 is the cost of lifejackets be the y-intercept.

Now using the slope-intercept form, we can have the equation by substituting the value of where m = 20 and b = 12, that is
y = mx + b
y = 20x + 12

The graph shows the activity in a savings account.

Question 12.
What was the amount of the initial deposit that started this savings account?
Answer:
Reading from the graph, the y-intercept is the starting point: (0, 1000), so the value for the y-intercept is 1000. Thus,
b = 1000$.
So that value is the amount of the initial deposit that he started with his savings account.

Lesson 5.1 Answer Key 8th Grade Writing Linear Equations Question 13.
Find the slope and y-intercept of the graphed line.
Answer:
From the given graph, find two points that will help you calculate the slope.
I’ll take (0, 1000) for (x₁, y₁) and (6, 4000) for (x₂, y₂).
Thus:
m = \(\frac{4000-1000}{6-0}\) = \(\frac{3000}{6}\) = 500
The starting point on the graph is (0, 1000), i.e. (0, b). So value of the y-intercept is 1000. → b = 1000

Question 14.
Write an equation in slope-intercept form for the activity in this savings account.
Answer:
The standard form of the linear equation is y = mx + b. In previous exercise we got values for the slope and y intercept → m = 500, b = 1000. Include the obtained values of the slope and y-intercept into the equation.
y = 500x + 1000

Let us describe the equation: y represents the total amount of saved money, x represents the time of collecting money (in months), the slope represents the money saved during x months and y-intercept represents the amount with which he started to save.

Question 15.
Explain the meaning of the slope in this graph.
Answer:
The slope represents the amount of money saved in dollars per month in plan.

Texas Go Math Grade 8 Lesson 5.1 H.O.T. Higher Order Thinking Answer Key 

Question 16.
Communicate Mathematical Ideas Explain how you decide which part of a problem will be represented by the variable x, and which part will be represented by the variable y in a graph of the situation.
Answer:
Explanation A:
You need to analyze the problem to find out which part depends on another. That dependent part will be represented by variable y, and the independent part by x.
You need to analyze the problem to find out which part depends on another.

Explanation B:
Variable x is the input, which represents the variable that affects the other variable when changed and is independent.
Variable y is the output, which represents the resulting variable or the variable that is dependent on the other.

Question 17.
Represent Real-World Problems Describe what would be true about the rate of change in a situation that could not be represented by a graphed line and an equation in the form y = mx + b.
Answer:
The rate of change would not be constant. Using different pairs of points in the slope formula would give different results.

Lesson 5.1 Writing Linear Equations from Situations and Graphs Answer Key Question 18.
Draw Conclusions Must m, in the equation y = mx + b, always be a positive number? Explain.
Answer:
In the equation y = mx + b, the variable m represents the slope of a line which is defined as the number that describes the steepness and direction of the line.

The slope of a line can be positive, negative, zero, or undefined.

• If the slope is upward to the right, then it is a positive slope.
• If the slope is downward to the right, then it is a negative slope.
• If the slope is horizontal, then the slope is zero.
• If the slope is vertical, then the slope is undefined.

We can conclude that m which is the slope of a line is not always a positive number as stated by the definition of the slope it can be a positive, negative, zero, or undefined.

Refer to our Texas Go Math Grade 8 Answer Key Pdf to score good marks in the exams. Test yourself by practicing the problems from Texas Go Math Grade 8 Lesson 6.2 Answer Key Describing Functions.

Texas Go Math Grade 8 Lesson 6.2 Answer Key Describing Functions

Essential Question
What are some characteristics that you can use to describe functions?

Texas Go Math Grade 8 Lesson 6.2 Explore Activity Answer Key

Investigating a Constant Rate of Change
The U.S. Department of Agriculture defines heavy rain as rain that falls at a rate of 1.5 centimeters per hour.
A. The table shows the total amount of rain that falls in various amounts of time during a heavy rain. Complete the table.

B. Plot the ordered pairs from the table on the coordinate plane at the right.
C. How much rain falls in 3.5 hours? ___
D. Plot the point corresponding to 3.5 hours of heavy rain.

E. What do you notice about all of the points you plotted?
F. Is the total amount of rain that falls a function of the number of hours that rain has been falling? Why or why not?

Reflect

Question 1.
Suppose you continued to plot points for times between those in the table, such as 1.2 hours or 4.5 hours. What can you say about the locations of these points?
Answer:
These points will lie on the straight line of the graph of the data.

Your Turn

Go Math Grade 8 Lesson 6.2 Answer Key Question 2.
State whether the relationship between x and y in y = 0.5x is proportional or nonproportional. Then graph the function.

Answer:
y = 0.5x Given

Proportional relationship Compare the equation with the general linear equation y = mx + b. Since b = 0, the relationship is proportional

Choose several values for the input x. Substitute these values of x in the equation to find the output y.

Graph the ordered pairs. Then draw a line through the points to represent the solution.

Your Turn

Question 3.
A soda machine makes \(\frac{2}{3}\) gallons of soda every minute. The total amount y that the machine makes in x minutes is given by the equation y = \(\frac{2}{3}\)x. Use the table and graph to determine whether the relationship between x and y is linear and, if so, if it is proportional.


Answer:
First, we compare the equation with the general linear equation y = mx + b. y = \(\frac{2}{3}\)x is in the form y = mx + b, with m = \(\frac{2}{3}\) and b = 0. Therefore the equation is a linear equation. Since b = 0, the relationship is proportional.
Then we choose several values for the input x. We substitute these values of x in the equation to find the output y.

We graph the ordered pairs and we draw a line through the points to represent the solutions of the function.

Texas Go Math Grade 8 Lesson 6.2 Guided Practice Answer Key

Plot the ordered pairs from the table. Then graph the function represented by the ordered pairs and tell whether the function is linear or nonlinear. Tell whether the function is proportional. (Examples 1 and 2)

Question 1.
y = 5 – 2x

Answer:
y = 5 – 2x

Complete the table

Graph the ordered pairs. Then draw a line through the points to represent the solution.
A linear relationship Graph of a Linear function is a straight-line

Lesson 6.2 Representing Functions Answer Key Question 2.
y = 2 – x²

Answer:
y = 2 – x² Given

Complete the table

Graph the ordered pairs. Then draw a line through the points to represent the solution.
Non linear relationship Graph of a Linear function is a straight line

Explain whether each equation is a linear equation. (Example 2)

Question 3.
y = x² – 1
Answer:
y = x² – 1
Insert several vaLues for X:
x = 2
y = 2² – 1 (Substitute the given value of x for x)
= 4 – 1 (Simplify)
= 3 (Subtract)
x = 3
y = 3² – 1 (Substitute the given value of x for x)
= 9 – 1 (Simplify)
= 8 (Subtract)
x = 4
y = 4² – 1 (Substitute the given value of x for x)
= 16 – 1 (Simplify)
= 15 (Subtract)
The rate of change is not constant, so the equation is non-linear.
Non-linear

Question 4.
y = 1 – x
Answer:
y = 1 – x Given
The equation is in the form of a linear equation, hence is a linear equation.
Compare the equation with the general linear equation y = mx + b.

Essential Question Check-In

Question 5.
Explain how you can use a table of values, an equation, and a graph to determine whether a function represents a proportional relationship.
Answer:
From a table, determine the ratio \(\frac{y}{x}\). If it is constant the relationship is proportional.

From a graph, note if the graph passes through the origin. The graph of the proportional relationship must pass through the origin (0, 0).

From an equation, compare with the general linear form of the equation, y = mx + b. If b = 0, the relationship is proportional.

Texas Go Math Grade 8 Lesson 6.2 Independent Practice Answer Key

Lesson 6.2 Answer Key 8th Grade Go Math Question 6.
State whether the relationship between x and y in y = 4x – 5 is proportional or nonproportional. Then graph the function.

Answer:
First, we compare the equation with the general linear equation y = mx + b. y = 4x – 5 is in the form y = mx + b, with m = 4 and b = -5. Therefore, the equation is a linear equation. Since b ≠ 0, the relationship is non-proportional.
Then, we choose several values for the input x. We substitute these values of x in the equation to find the output y.

We graph the ordered pairs and we draw a line through the points to represent the solutions of the function.

Go Math Lesson 6.2 Describing Function Relationships Answer Key Question 7.
The Fortaleza telescope in Brazil is a radio telescope. Its shape can be approximated with the equation y = 0.01 3x². Is the relationship between x and y linear? Is it proportional? Explain.
Answer:
The linear function has the form y = mx + b, where m and b are real numbers. Every equation in the form of y = mx + b is a linear equation. Equations that cannot be written in this form is not linear equations and not linear functions.

Given the equation y = 0.013x², we can definitely say that the relationship of x and y is not linear because the equation is not in the form of y = mx + b. As we can see, in the given equation there is variable x that is being raised to the 2 power or simply x² which is not noticeable in the form of linear equation y = mx + b. So, it is not linear.

Since the given is not a linear equation, we cannot identify having a proportional or nonproportional relationship, then the equation is not proportional

Question 8.
Kiley spent $20 on rides and snacks at the state fair. If x is the amount she spent on rides, and y is the amount she spent on snacks, the total amount she spent can be represented by the equation x + y = 20. Is the relationship between x and y linear? Is it proportional? Explain.
Answer:
Owen
x + y = 20 Given
y = -x + 20 Rewriting the equation
It is linear Compare the equation with the general linear equation y = mx + b.
It is not proportional Since b ≠ 0, the relationship is not proportionaL

Question 9.
Represent Real-World Problems The drill team is buying new uniforms. The table shows y, the total cost in dollars, and x, the number of uniforms purchased.

a. Use the data to draw a graph. Is the relationship between x and y linear? Explain.
Answer:

Draw the graph
x and y are linear Graph of a linear relationship is a straight line.

b. Use your graph to predict the cost of purchasing 12 uniforms.
Answer:
The cost of 12 uniforms is $720

Go Math 8th Grade Answer Key Pdf Lesson 6.2 Reteach Answer Key Question 10.
Marta, a whale calf in an aquarium, is fed a special milk formula. Her handler uses a graph to track the number of gallons of formula y the calf drinks in x hours. Is the relationship between x and y linear? Is it proportional? Explain.

Answer:
The relationship is linear The data lies on a straight line

The relationship is proportional. The graph passes through the origin

Question 11.
Critique Reasoning A student claims that the equation y = 7 is not a linear equation because it does not have the form y = mx + b. Do you agree or disagree? Why?
Answer:
Disagree The equation can be written in the form y = mx + b where m is 0. The graph of the solutions is a horizontal line.

Question 12.
Make a Prediction Let x represent the number of hours you read a book and y represent the total number of pages you have read. You have already read 70 pages and can read 30 pages per hour. Write an equation relating x hours and y pages you read. Then predict the total number of pages you will have read after another 3 hours.
Answer:
In the problem, the number of pages you have read already which is 70 pages and you can read 30 pages per hour, let us determine the equation relating x hours and y pages you read. Also, let us predict the total number of pages you will have read after another 3 hours.

Representation:
Let x be the number of hours you read a book
y be the number of pages you have read
m be the number of pages per x hours
b be the number of pages you have read already

To determine the equation, let us apply the slope-intercept form y = mx + b. By using the representation, we can
formulate the equation given m = 30 and b = 70, that is y = 30x + 70

Now, Let us predict the total number of pages you will have read after another 3 hours. Using the equation y =
30x + 70, substitute the 3 as the value of x.
y = 30x + 70
y = 30(3) + 70
y = 90 + 70
y = 160

Thus, the number of pages you will have read after another 3 hours is 160 pages.
y = 30x + 70; 160 pages

Texas Go Math Grade 8 Lesson 6.2 H.O.T. Focus On Higher Order Thinking Answer Key

Lesson 6.2 Describing Functions Answer Key Go Math 8th Grade Question 13.
Draw Conclusions Rebecca draws a graph of a real-world relationship that turns out to be a set of unconnected points. Can the relationship be linear? Can it be proportional? Explain your reasoning.
Answer:
The relationship is linear if all, the points lie on the same line.

If the relationship is linear and passes through the origin, it is proportional.

Question 14.
Communicate Mathematical Ideas Write a real-world problem involving a proportional relationship. Explain how you know the relationship is proportional.
Answer:
The amount of money earned at a car wash is a proportional relationship. When there is 0 cars washed, $0 is earned. The amount of money earned increases by the unit cost of car wash.

Go Math 8th Grade Lesson 6.2 Answer Key Describing Functions Question 15.
Justify Reasoning Show that the equation y + 3 = 3(2x + 1) is linear and that it represents a proportional relationship between x and y.
Answer:
y + 3 = 3(2x + 1) Given
y + 3 = 6x + 3 Simplify using distributive property
y + 3 – 3 = 6x + 3 – 3 Subtract 3 from each side
y = 6x
Compared with the general linear equation y = mx + b
Proportional Since b = 0, it is a proportional relationship

Refer to our Texas Go Math Grade 8 Answer Key Pdf to score good marks in the exams. Test yourself by practicing the problems from Texas Go Math Grade 8 Lesson 4.1 Answer Key Representing Linear Nonproportional Relationships.

Texas Go Math Grade 8 Lesson 4.1 Answer Key Representing Linear Nonproportional Relationships

Essential Question
How can you use tables, graphs, and equations to represent linear nonproportional situations?

Your Turn

Question 1.
Francisco makes $12 per hour doing part-time work on Saturdays. He spends $4 on transportation to and from work. The equation y = 12x – 4 gives his earnings y, after transportation costs, for working x hours. Make
a table of values for this situation.

Answer:
Given equation y = 12x – 4.
Choose several values for x z and substitute in the equation.
Make a table.

How did we get values for y:
For x = 1 : y = 12 ∙ 1 – 4 = 8
For x = 2 : y = 12 ∙ 2 – 4 = 20
For x = 3: y = 12 ∙ 3 – 4 = 32
For x = 4 : y = 12 ∙ 4 – 4 = 44

Choose several values for x and substitute in the equation. Make a table.

Texas Go Math Grade 8 Lesson 4.1 Explore Activity Answer Key 

Examining Linear Relationships
Recall that a proportional relationship is a relationship between two quantities in which the ratio of one quantity to the other quantity is constant. The graph of a proportional relationship is a line through the origin. Relationships can have a constant rate of change but not be proportional.

The entrance fee for Mountain World theme park is $20. Visitors purchase additional $2 tickets for rides, games, and food. The equation y = 2x +20 gives the total cost, y, to visit the park, including purchasing x tickets.

Step 1
Complete these exercises to review skills you will need for this chapter.

Step 2
Plot the ordered pairs from the table. Describe the shape of the graph.
Step 3
Find the rate of change between each point and the next. Is the rate constant?
Step 4
Calculate \(\frac{y}{x}\) for the values in the table. Explain why the relationship between number of tickets and total cost is not proportional.

Reflect

Lesson 4.1 Go Math Grade 8 Answer Key Pdf Question 2.
Analyze Relationships Would it make sense to add more points to the graph from x = 0 to x = 10? Would it make sense to connect the points with a line? Explain.
Answer:
The given equation is y = 2x + 20, which shows the proportion of the total cost and the number of additional
tickets.

To get the values for y, include the given values of x in the given equation.

After entering the points on the graph, we get a straight line, which does not pass through the origin.

To find the rate of change between each point and the next one, we need to get the slope of the line. The
standard form of the given equation is y = mx + b, so we are calculating the m (slope).
m = \(\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\)
Look at the dots from the graph:

The rate is constant because the ratio between each and the next point is the same.

In the previous step, we have calculated the rate of change. The rate is constant, but the linear relationship is not proportional because the line does not pass through the origin.

Take a look at the standard form of the equation again: y = mx + b. when the b is not equal to 0, b ≠ 0, the linear
function won’t pass the origin on the graph.

Your Turn

Question 3.
Make a table and graph the solutions of the equation y = -2x + 1.


Answer:
y = -2x + 1.

Choose several values for x and substitute in the equation to find y. Make a table.

Plot the ordered pairs from the table. Then draw a line connecting the points to represent all the possible solutions.

Texas Go Math Grade 8 Lesson 4.1 Guided Practice Answer Key 

Make a table of values for each equation. (Example 1)

Question 1.
y = 2x + 5

Answer:
For the given equation y = 2x + 5, defaulted are values for x. To find the values for y, include the given values of x in the given equation.

Go Math Grade 8 Lesson 4.1 Answer Key Question 2.
y = \(\frac{3}{8}\)x – 5

Answer:
y = \(\frac{3}{8}\)x – 5 Given

Choose several values for x and substitute in the equation to find y. Make a table.

Explain why each relationship is not proportional. (Explore Activity)

Question 3.

First calculate \(\frac{y}{x}\) for the values in the table.
Answer:
\(\frac{3}{0}\) = undefined Find \(\frac{y}{x}\)
\(\frac{7}{2}\) = 3.5
\(\frac{11}{4}\) = 2.75
\(\frac{15}{6}\) = 2.5
\(\frac{19}{8}\) = 2.375
The ratio is not constant, hence relationship is not proportional.

Question 4.

Answer:
The graph is a straight line but does not pass through the origin.

Complete the table for the equation. Then use the table to graph the equation. (Example 2)

Question 5.
y = x – 1


Answer:
y = x – 1 Given
Choose several values for x and substitute in the equation to find y. Make a table.

Plot the ordered pairs from the table. Then draw a line connecting the points to represent all the possible solutions.

Essential Question Check-In

Question 6.
How can you choose values for x when making a table of values representing a real-world situation?
Answer:
When choosing values for x in a real-world situation, you choose positive values with an appropriate interval to represent the array of data.

Texas Go Math Grade 8 Lesson 4.1 Independent Practice Answer Key 

State whether the graph of each linear relationship is a solid line or a set of unconnected points. Explain your reasoning.

Question 7.
The relationship between the number of $4 lunches you buy with a $100 school lunch card and the money remaining on the card
Answer:
Each time you buy lunch the amount on the card will decrease by 4. This means that this function will be y = 100 – 4x, where x is the number of lunches and y remaining money.
For example:
When we buy one lunch (x = 1), we have 96 left.
That way we can only buy 25 lunches for 100.
The number x must be an integer because we cannot buy a fractional part of a lunch. So we have a set of unconnected points:{( 1, 96), (2, 92), …, (25, 0)}, not a solid line.

Set of unconnected points

Texas Go Math Lesson 4.1 Answer Key Grade 8 Question 8.
The relationship between time and the distance remaining on a 3-mile walk for someone walking at a steady
rate of 2 miles per hour
Answer:
The person who is going to walk starts from the starting point 0. Thus, the first point on the graph is (0, 0), the starting point. The second thing we know is that a person has crossed 2 miles in an hour, so that’s another point on the graph (1, 2).
Those data are based on the rate of steady walking.
Since the rate of steady walking is the same all the time, it is constant, and time can be measured in fraction, we can read the value for x on 3 miles (y) from the graph. → x = 1.5 = \(\frac{3}{2}\)

The state of this graph of the linear relationship is a solid line.

Question 9.
Analyze Relationships Simone paid $12 for an initial year’s subscription to a magazine. The renewal rate is $8 per year. This situation can be represented by the equation y = 8x + 12, where x represents the number of years the subscription is renewed and y represents the total cost.
a. Make a table of values for this situation.

Answer:
y = 8x + 12
to make a table, we need to take some values for x, then include them into the equation y = 8x + 12 to get the values for y.

b. Draw a graph to represent the situation.
Answer:
enter to the graph obtained values of x and y

c) Explain why this relationship is not proportional.
Answer:
When the value of b is not equal to 0, b ≠ 0. the linear function does not pass through the origin. If the line does not pass the origin, the linear function is NON-proportional.
In our case b = 12 so it is not proportional.

d. Does it make sense to connect the points on the graph with a solid line? Explain.
Answer:
If we were to connect the obtained points on the graph, we would get a straight line, but that makes no sense.
The subscription works in a way that it is not possible to pay any amount other than $ 8 per year and it is not
possible to pay the subscription amount for just half a year or a few months. Any other point on the graph would
not be an actual indicator of the total cost.

Lesson 4.1 Representing Linear Non-Proportional Relationships Answer Key Question 10.
Analyze Relationships A proportional relationship is a linear relationship because the rate of change is constant (and equal to the constant of proportionality). What is required of a proportional relationship that is not required of a general linear relationship?
Answer:
All proportional relationships are linear relationships, but all linear relationships are not proportional relationships. So, the only difference between those is b from the equation y = mx + b. The linear function is proportional if the b = 0 and passes through the Origin. If b ≠ 0, then it is non-proportional, but it is still linear.

Question 11.
Communicate Mathematical Ideas Explain how you can identify a linear nonproportional relationship from a table, a graph, and an equation.
Answer:
In a table, the ratios \(\frac{y}{x}\) will not be equal.
A graph will not pass through the origin.
An equation will be in the form y = mx + b where b ≠ 0

Texas Go Math Grade 8 Lesson 4.1 H.O.T. Focus On Higher Order Thinking Answer Key 

Question 12.
Critique Reasoning George observes that for every increase of 1 in the value of x, there is an increase of 60 in the corresponding value of y. He claims that the relationship represented by the table is proportional. Critique George’s reasoning.

Answer:
\(\frac{90}{1}\) = 90 Find \(\frac{y}{x}\)
\(\frac{150}{2}\) = 75
\(\frac{210}{3}\) = 70
\(\frac{270}{4}\) = 67.5
\(\frac{330}{5}\) = 66
The ratio is not constant, hence the relationship cannot be proportional.

Go Math Middle School Grade 8 Answer Key Pdf Lesson 4.1 Reteach Answer Key Question 13.
Make a Conjecture Two parallel lines are graphed on a coordinate plane. How many of the lines could represent proportional relationships? Explain.
Answer:
The linear function can be proportional just if b = 0, and passes through the origin. If there are two lines on the
graph, just one can pass through the origin. Thus, if one line passes through the origin, the other one won’t, so just the ONE line can represent the proportional relationship.

Refer to our Texas Go Math Grade 8 Answer Key Pdf to score good marks in the exams. Test yourself by practicing the problems from Texas Go Math Grade 8 Lesson Texas Go Math Grade 8 Lesson 6.1 Answer Key Identifying and Representing Functions.

Texas Go Math Grade 8 Lesson 6.1 Answer Key Identifying and Representing Functions

Essential Question
How can you identify and represent functions?

Texas Go Math Grade 8 Lesson 6.1 Explore Activity Answer Key 

Explore Activity
Understanding Relationships
Carlos needs to buy some new pencils from the school supply store at his school. Carlos asks his classmates if they know how much pencils cost. Angela says she bought 2 pencils for $0.50. Paige bought 3 pencils for $0.75, and Spencer bought 4 pencils for $1.00.

Carlos thinks about the rule for the price of a pencil as a machine. When he puts the number of pencils he wants to buy into the machine, the machine applies a rule and tells him the total cost of that number of pencils.

A. Use the prices in the problem to fill in total cost in rows i – iii of the table.

B. Describe any patterns you see. Use your pattern to determine the cost of 1 pencil.

C. Use the pattern you identified to write the rule applied by the machine. Write the rule as an algebraic expression and fill in rule column row iv of the table.

D. Carlos wants to buy 12 pencils. Use your rule to fill in row v of the table to show how much Carlos will pay for 12 pencils.

Reflect

Question 1.
How did you decide what operation to use in your rule?
Answer:
The rule would be to multiply the number of pencils to the unit rate to find the total cost.

Identifying and Representing Functions Lesson 6.1 Answer Key Question 2.
What If? Carlos decides to buy erasers in a package. There are 6 pencil-top erasers in 2 packages of erasers.
Answer:
To write the rule as an algebraic expression, we will denote the number of pencil-top erasers with variables y and x will represent the package of erasers.

a. Write a rule in words for the number of packages Carlos needs to buy to get x erasers. Then write the rule as an algebraic expression.
Answer:
Knowing that Carlos got six pencil-top erasers in two packages, we can conclude that one package results in
three erasers, so we can write it down as:
x = \(\frac{y}{3}\)
Therefore, we can claim that for every bought package, Carlos gets three pencil-top erasers.

b. How many packages does Carlos need to buy to get 18 erasers?
Answer:
To calculate how many packages Carlos has to buy to get 18 erasers, we will use the given formula as following:
x = \(\frac{18}{3}[/latex
x = 6

Reflect

Question 3.
Is it possible for a function to have more than one input value but only one output value? Provide an illustration to support your answer.

Answer:
We know that a function assigns exactly one output to each input. Therefore, it is possible for a function to have more than one input value but only one output value. Below, you can find an illustration of this case.

Your Turn

Determine whether each relationship is a function. Explain.

Question 4.

Answer:
Function A function assigns exactly one output to each input.

Identifying and Representing Functions Answer Key Question 5.

Answer:
Not function A function assigns exactly one output to each input.

Reflect

Question 6.
What is always true about the numbers in the first column of a table that represents a function? Why must this be true?
Answer:
To determine the answer here, we will review the definition of a function.

In the first column of a table that represents a function, there should be no repetition of values. By the definition, a table of values represents a function only if each input value has one output value.
Therefore, two values in one column would mean that the table does not represent a function.

Your Turn

Determine whether each relationship is a function. Explain.

Question 7.

Answer:
Function There is no repetition in the first column. Each input value is paired with only one output value.

Question 8.

Answer:
Not a function There is a repetition in the first column which is 8. Hence, 8 has two outputs 21 and 34.

Question 9.
Many real-world relationships are functions. For example, the amount of money made at a car wash is a function of the number of cars washed. Give another example of a real-world function.
Answer:
Profit at the coffee shop can be viewed as a function, considering the fact that it depends on the other variable which is the amount of sold products.

Therefore, we can say that the profit is a function of the number of sold products

Your Turn

Lesson 6.1 Identifying and Representing Functions Answer Key Question 10.
The graph shows the relationship between the heights and weights of the members of Team Members of a basketball team. Is the relationship represented by the graph a function? Explain.

Answer:
Not a function Input values are paired with more than one output values (70, 165) and (70, 174).

Texas Go Math Grade 8 Lesson 6.1 Guided Practice Answer Key

Complete each table. In the row with x as the input, write a rule as an algebraic expression for the output. Then complete the last row of the table using the rule. (Explore Activity)

Question 1.

Answer:
Unit cost of ticket = [latex]\frac{40}{2}\) = 20

Total Cost = 20x
where $x$ is the number of tickets

Question 2.

Answer:
Number of pages per minute = \(\frac{1}{2}\) = 0.5
Total Cost = 0.5x
where $x$ is the number of minutes

Question 3.

Answer:
Unit cost of Muffins = \(\frac{2.25}{1}\) = 2.25
Total Cost = 2.25x
where $x$ is the number of muffins

Determine whether each relationship is a function. (Examples 1 and 2)

Go Math Lesson 6.1 Identifying and Representing Functions Answer Key Question 4.

Answer:
Function Each input is assigned to exactly one output

Question 5.

Answer:
To determine the answer here, we will review the definition of a function.

If a certain table of values represents a function, there should be no repetition of values in the first column because in a function each input value is paired with one output value.

Therefore, the given relationship does not represent a function, because value 4 repeats two times in the first column and results in two output values.

Question 6.
The graph shows the relationship between the weights of 5 packages and the shipping charge for each package. Is the relationship represented by the graph a function? Explain.

Answer:
Explanation A:
Because inputs are paired with output value and there is no repeated value, the relationship is a function.

Explanation B:
To determine the answer, we will write values from the graph in one table. First column represents the weight and the second one represents the shipping cost. Table is shown below.

Considering the fact that there is no repetition of input values and every input value is paired with one output value, we can claim that given relationship is a function.

Essential Question Check-In

Question 7.
What are four different ways of representing functions? How can you tell if a relationship is a function?
Answer:
A function is defined as the relationship in which the input value is paired with exactly one output value. In representing functions, we have four different ways. These are using Mapping Diagrams, Tables, Graphs, and Equations.

A mapping diagram represents a function if each input value is paired with only one output value.

Relationships of functions between input values and output values can also be represented using tables. The values in the first column are the input values. The values in the second column are the output values.

Graphs can be used to displays relationships between two sets of numbers. Each point on a graph represents an ordered pair

A function can be represented by an equation with two variables. One variable is the domain and the other is the
range.

By the description, the four ways of representing a function are through Mapping Diagrams, Tables, Graphs, and
Equations Also, we can say that a relationship is a function if the input value is paired with exactly one output value.

Texas Go Math Grade 8 Lesson 6.1 Independent Practice Answer Key

Determine whether each relationship represented by the ordered pairs is a function. Explain.

Question 8.
(2, 2), (3, 1), (5, 7), (8, 0), (9, 1)
Answer:
A function is defined as a relation such that the input value is paired with exactly one output value.

To determine if the given ordered pairs (2, 2), (3, 1), (5, 7), (8, 0), (9, 1) is a function, we can write it in table form, wherein x-coordinate is represented as input and y-coordinate is represented as output.
|input (x) | Output(y)|
|——|——|
|2 |2|
|3|1|
|5 | 7|
|8| 0|
| 0|1|

As we can notice in the table, one output value is paired with two input values these are (3, 1) and (0, 1), however
each input value is paired only with one output value, so still, the relationship is a function.

A function; Each input value is paired only with one output value.

Lesson 6.1 Answer Key 8th Grade Representing Functions Answer Key Question 9.
(0, 4), (5, 1), (2, 8), (6, 3), (5, 9)
Answer:
A function is defined as a relation such that the input value is paired with exactly one output value.

To determine if the given ordered pairs (0,1), (5, 1), (2,8), (6,3), (5,9) is a function, we can write it in table form, wherein the x-coordinate is represented as input and the y-coordinate is represented as output
|Input (x) | Output (y)l
|–|–|
|0|4|
|5|1|
|2 | 8|
|6| 3|
|5| 9|

As we can notice in the table, one input value “5” is paired with two output values “1 and 9”, (5, 1) and (5, 9), so the relationship is not a function.
Not a function; The input value 5 is paired with more than one output value 1 and 9.

Question 10.
Draw Conclusions Joaquin receives $0.40 per pound for 1 to 99 pounds of aluminum cans he recycles. He receives $0.50 per pound if he recycles more than 100 pounds. Is the amount of money Joaquin receives a function of the weight of the cans he recycles? Explain your reasoning.
Answer:
Yes The amount of money increases with the weight of the cans. No weight will result in the same amount of money earned.

Question 11.
A biologist tracked the growth of a strain of bacteria, as shown in the graph.

a. Explain why the relationship represented by the graph is a function.
Answer:
The relationship is a function as each input has been assigned exactly one output. There is only one number of bacteria for each number of hours.

b. What is It? Suppose there was the same number of bacteria for two consecutive hours. Would the graph still represent a function? Explain.
Answer:
Yes. If the number of bacteria for two consecutive hours is the same, one input will still be paired with one output hence the relationship is still a function.

Question 12.
Multiple Representations Give an example of a function in everyday life and represent it as a graph, a table, and a set of ordered pairs. Describe how you know it is a function.


Answer:
The cost of a bouquet of flowers and the number of flowers in the bouquet is a function. Unit cost of flowers = $0.85 and x is the number of flowers. Hence. C = 0.85x


Ordered Pairs
(2, 1.7), (4, 3.4), (6, 5.1), (8, 6.8), (10, 8.5)
Each value of input is paired with exactly one output.

The graph shows the relationship between the weights of six wedges of cheese and the price of each wedge.

Lesson 6.1 Identifying and Representing Functions Go Math Question 13.
Is the relationship represented by the graph a function? Justify your reasoning. Use the words ‘input” and output” in your explanation, and connect them to the context represented by the graph.

Answer:
Yes, the relationship represented by the graph is a function. Each input (weight) in the graph is paired with exactly one output (price).

Yes, the relationship represented by the graph is a function.

Question 14.
Analyze Relationships Suppose the weights and prices of additional wedges of cheese were plotted on the graph. Is that likely to change your answer to question 13? Explain your reasoning.

Answer:
Explanation A:
No, since the weight of the cheese is directly proportional to the cost of wedges of cheese.

Explanation B:
If weights and prices of additional wedges of cheese were plotted on the graph, it would still result in a function

The relationship would still result in a function because the price of wedges and weight are proportional values, meaning that for every input (weight) there would be only one output (price).

Texas Go Math Grade 8 Lesson 6.1 H.O.T Focus On Higher Order Thinking Answer Key

Question 15.
Justify Reasoning A mapping diagram represents a relationship that contains three different input values and four different output values. Is the relationship a function? Explain your reasoning.
Answer:
No. Since there are three inputs and four outputs, one of the inputs will have more than one output, hence the relationship cannot be a function.

Question 16.
Communicate Mathematical Ideas An onion farmer is hiring workers to help harvest the onions. He knows that the number of days it will take to harvest the onions is a function of the number of workers he hires. Explain the use of the word “function” in this context.
Answer:
In the problem, An onion farmer is hiring workers to help harvest the onions. He knows that the number of days it will take to harvest the onions is a function of the number of workers he hires. The concept of the function is being applied in the context.

The word function in the context of the problem tells that the input or the independent variable x is the number of workers the onion farmers hire on the farm, while the output or the dependent variable y is the number of days it will take to harvest the onions.

The function in the problem is used to describe that the input will greatly affect the output since the more workers hired will result in fewer days of harvest As x increases, y decreases or vice versa. In relation to function, each worker has a unique element or contribution to the output