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) = (y_{2} – y_{1}) ÷ (x_{2} – x_{1})

\(\frac{52-42}{8-6}\) = 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

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 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 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 into 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

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

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 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 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**

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 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 = \(\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\)

First point is (1, 125) and the second point is (2, 175). Put those in 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

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) = (y_{2} – y_{2}) ÷ (x_{2} – x_{1})

where

(x_{2}, y_{2}) = (2, 29) and (x_{1}, y_{1}) = (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) = (y_{2} – y_{1}) ÷ (x_{2} – x_{1})

Rate of change is $\$30$ per lesson

where (x_{2}, y_{2}) = (2, 85) and (x_{1}, y_{1}) = (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) = (y_{2} – y_{1}) ÷ (x_{2} – x_{1})

Rate of change is $\$50$ per lesson

where (x_{2}, y_{2}) = (2, 125) and (x_{1}, y_{1}) = (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 same. Comparing results from part a and b.

Rate of change is higher for private lessons then group lessons.

**Vocabulary Explain why each relationship is not linear.**

Question 11.

Answer:

\(\frac{6.5-4.5}{2-1}\) Finding the rate of change using given points by

slope (m) = (y_{2} – y_{1}) ÷ (x_{2} – x_{1})

\(\frac{8.5-6.5}{3-2}\) = 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 buy 5 additional cards every month. Real world situation

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 earning become $600 double of his initial base salary of $300. Hence, after parking 61 cars, his earning from fee becomes more than his fixed salary.

\(\frac{300}{5}\) = 60

He earns the same ¡n fees as his fixed salary for parking.