? ESSENTIAL QUESTION

Writing an Equation in Slope-Intercept FormGreta 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.

What is the input variable, x, for this situation?

What is the output variable, y, for this situation?

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?

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?

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

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

L E S S O N

5.1Writing Linear Equations from Situations and Graphs

EXPLORE ACTIVITY

What change could the studio make that would make a difference to the

y-intercept of the equation?

Math TalkMathematical Practices

What change could the

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8.F.2.4

Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value… . Interpret the rate of change and initial value… .

8.F.2.4

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Writing an Equation from a GraphYou can use information presented in a graph to write an equation in slope-intercept form.

A video club charges a one-time membership fee plus a rental fee for each DVD borrowed. Use the graph to write an equation in slope-intercept form to represent the amount spent, y, on x DVD rentals.

Choose two points on the graph, (x1, y1) and (x2, y2), to find the slope.

m = y2 - y1 ______ x2 - x1

m = 18 - 8 ______ 8 - 0

m = 10 ___ 8 = 1.25

Read the y-intercept from the graph.

The y-intercept is 8.

Use your slope and y-intercept values to write an equation in slope-intercept form.

y = mx + b

y = 1.25x + 8

Reflect 1. What does the value of the slope represent in this context?

2. Describe the meaning of the y-intercept.

EXAMPLE 1

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STEP 2

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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.

YOUR TURN

If the graph of an equation is a line that goes through the

origin, what is the value of the y-intercept?

Math TalkMathematical Practices

If the graph of an equation

Math Talk

8.F.2.4

Find the change in y-values over the change in x-values.Substitute (0, 8) for (x1, y1) and (8,18) for (x2, y2).

Simplify.

Slope-intercept form

Substitute 1.25 for m and 8 for y.

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My Notes

600 square feet for $750900 square feet for $1150

West Main St. O�ce RentalsO�ces for rent at convenient locations.

Monthly Rates:

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Writing an Equation from a DescriptionYou can use information from a description of a linear relationship to find the slope and y-intercept and to write an equation.

The rent charged for space in an office building is a linear relationship related to the size of the space rented. Write an equation in slope-intercept form for the rent at West Main Street Office Rentals.

Identify the input and output variables.

The input variable is the square footage of floor space.

The output variable is the monthly rent.

Write the information given in the problem as ordered pairs.

The rent for 600 square feet of floor space is $750: (600, 750)

The rent for 900 square feet of floor space is $1150: (900, 1150)

Find the slope.

Find the y-intercept. Use the slope and one of the ordered pairs.

y = mx + b

750 = 4 _ 3 · 600 + b

750 = 800 + b

-50 = b

Substitute the slope and y-intercept.

y = mx + b

y = 4 _ 3 x - 50

Reflect 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?

EXAMPLEXAMPLE 2

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m = y2 - y1 ______ x2 - x1

= 1150 - 750 __________ 900 - 600 = 400 ____ 300 = 4 __ 3

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STEP 5

8.F.2.4

Slope-intercept form

Substitute for y, m, and x.

Multiply.

Subtract 800 from both sides.

Slope-intercept form

Substitute 4 __ 3 for m and -50 for b.

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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.

YOUR TURN

Guided Practice

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:

b. output variable:

c. equation:

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 = y2 - y1 _____ x2 - x1

=

Read the y-intercept from the graph: b =

Use your slope and y-intercept values to write an equation in slope-intercept form.

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)

4. Explain what m and b in the equation y = mx + b tell you about the graph of the line with that equation.

ESSENTIAL QUESTION CHECK-IN???

Input variable: Output variable:

m = y2 - y1 _____ x2 - x1

= Use the slope and one of the ordered

pairs in y = mx + b to find b. = · + b; = b

Write an equation in slope-intercept form.

Unit 2130

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Independent Practice5.1

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.

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.

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

7. Use the graph to find the slope of the line. Tell what the slope means in this context.

8. Identify the y-intercept. Tell what the y-intercept means in this context.

9. Write an equation in slope-intercept form that represents the diver’s depth over time.

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.

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.

8.F.2.4

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The graph shows the activity in a savings account.

12. What was the amount of the initial deposit that started this savings account?

13. Find the slope and y-intercept of the graphed line.

14. Write an equation in slope-intercept form for the activity in this savings account.

15. Explain the meaning of the slope in this graph.

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.

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.

18. Draw Conclusions Must m, in the equation y = mx + b, always be a positive number? Explain.

FOCUS ON HIGHER ORDER THINKING

Unit 2132

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