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

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)
Texas Go Math Grade 8 Lesson 4.2 Answer Key 1
Texas Go Math Grade 8 Lesson 4.2 Answer Key 2

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.
Texas Go Math Grade 8 Lesson 4.2 Answer Key 3
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.
Texas Go Math Grade 8 Lesson 4.2 Answer Key 4
Answer:
The slope is calculated by:
m = \(\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\)
Thus:
Texas Go Math Grade 8 Lesson 4.2 Answer Key 19
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.
Texas Go Math Grade 8 Lesson 4.2 Answer Key 5

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.
Texas Go Math Grade 8 Lesson 4.2 Answer Key 6

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.
Texas Go Math Grade 8 Lesson 4.2 Answer Key 7
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.
Texas Go Math Grade 8 Lesson 4.2 Answer Key 8
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.
Texas Go Math Grade 8 Lesson 4.2 Answer Key 9
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.
Texas Go Math Grade 8 Lesson 4.2 Answer Key 10
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.
Texas Go Math Grade 8 Lesson 4.2 Answer Key 11
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:
Texas Go Math Grade 8 Lesson 4.2 Answer Key 20
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.
Texas Go Math Grade 8 Lesson 4.2 Answer Key 12
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:
Texas Go Math Grade 8 Lesson 4.2 Answer Key 21
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.
Texas Go Math Grade 8 Lesson 4.2 Answer Key 13
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 $\$5$

Texas Go Math Grade 8 Lesson 4.2 Answer Key 14

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.
Texas Go Math Grade 8 Lesson 4.2 Answer Key 15
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.
Texas Go Math Grade 8 Lesson 4.2 Answer Key 16
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.
Texas Go Math Grade 8 Lesson 4.2 Answer Key 17
Answer:
Calculate the slope for each and next point.
Texas Go Math Grade 8 Lesson 4.2 Answer Key 22

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.
Texas Go Math Grade 8 Lesson 4.2 Answer Key 18
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.

Leave a Comment

Scroll to Top
Scroll to Top