LESSON

6.1

Name

Slope-Intercept Form

Class

6.1

Date

Slope-Intercept Form

Essential Question: How can you represent a linear function in a way that reveals its slope and y-intercept?

Common Core Math Standards The student is expected to: COMMON CORE

Resource Locker

F-IF.C.7a

Graph linear... functions and show intercepts... Also A-CED.A.2, A-REI.D.10

Explore

Mathematical Practices COMMON CORE

Graphing Lines Given Slope and y-intercept

Graphs of linear equations can be used to model many real-life situations. Given the slope and y-intercept, you can graph the line, and use the graph to answer questions.

MP.6 Precision

Andrew wants to buy a smart phone that costs $500. His parents will pay for the phone, and Andrew will pay them $50 each month until the entire amount is repaid. The loan repayment represents a linear situation in which the amount y that Andrew owes his parents is dependent on the number x of payments he has made.

Language Objective Explain to a partner how to write a linear function in slope-intercept form.



When x = 0, y = $500 . The y-intercept of the graph of the equation that represents the situation is 500

You can determine the slope m of the graph of the function and its y-intercept b and write the equation y = mx + b, called the slope-intercept form of the equation.

PREVIEW: LESSON PERFORMANCE TASK View the Engage section online. Discuss the photo and how a gym membership may require a one-time sign-up fee as well as regular monthly fees. Also discuss how a graph of this type of data might look. Then preview the Lesson Performance Task.

The slope is

–50 .



Use the y-intercept to plot a point on the graph of the equation. The y-intercept is 500 ,



Using the definition of slope, plot a second point.

so plot the point (0, 500) .

–50 Change in y Slope = __ = _ = –50 . 1 Change in x Start at the point you plotted. Count 50 and



1

Amount Andrew Owes

units down

unit right and plot another point.

Amount ($)

Essential Question: How can you represent a linear function in a way that reveals its slope and y-intercept?

The rate of change in the amount Andrew owes over time is –$50 per month.

© Houghton Mifflin Harcourt Publishing Company • Image Credits: ©Echo/ Cultura/Getty Images

ENGAGE



Draw a line through the points you plotted.

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A1_MNLESE368170_U3M06L1 239

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Lesson 1 239 Module 6

L1 239

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A1_MNLE

Lesson 6.1

x 1 2 3 4 5 6 7 8 9 Time (Months)

Name

239

500 450 400 350 300 250 200 150 100 50

y

3/21/14

2:43 AM

3/21/14 3:25 AM

Reflect

EXPLORE

Discussion How can you use the same method to find two more points on that same line? Possible answer: You can begin at the second point, (1, 450), and move 50 units down and

1.

Graphing Lines Given Slope and y-Intercept

1 unit to the right. Then repeat this process beginning at the new point. How many months will it take Andrew to pay off his loan? Explain your answer. 10 months; the point (10, 0) represents the number of months, 10, for which the amount

2.

to be repaid is $0.

Explain 1

INTEGRATE TECHNOLOGY

Creating Linear Equations in Slope-Intercept Form

Students have the option of completing the activity either in the book or online.

You can use the slope formula to derive the slope-intercept form of a linear equation. Consider a line with slope m and y-intercept b. y2 - y1 The slope formula is m = _ x2 - x1 .

CONNECT VOCABULARY

Substitute (0, b) for (x 1, y 1) and (x, y) for (x 2, y 2). y-b m=_ x-0 y-b _ m= x

Remind students that the word intercept means to come together. When a player intercepts a football, the player and football come together at a certain point. Help students make the connection to the y-intercept on a graph, the place where the line “comes together” with the y-axis.

Multiply both sides by x (x ≠ 0).

mx = y - b mx + b = y

Add b to both sides.

y = mx + b

Slope-Intercept Form of an Equation If a line has slope m and y-intercept (0, b), then the line is described by the equation y = mx + b. Example 1

Slope is 3, and (2, 5) is on the line. Step 1: Find the y-intercept. y = mx + b 5 = 3(2) + b 5=6+b 5-6=6+b-6 -1 = b

Step 2: Write the equation. y = mx + b y = 3x + (-1) y = 3x - 1

Module 6

© Houghton Mifflin Harcourt Publishing Company



EXPLAIN 1

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

Write the slope–intercept form. Substitute 3 for m, 2 for x, and 5 for y. Multiply. Subtract 6 from both sides. Simplify. Write the slope–intercept form. Substitute 3 for m and -1 for b.

240

Creating Linear Equations in Slope-Intercept Form AVOID COMMON ERRORS Some students may not understand how to use the coordinates (x 1, y 1) and (x 2, y 2) to calculate the slope. Explain that the subscripts show which x-value goes with which y-value; for example the x-value of the first point is x 1, the y-value of the second point is y 2. Remind students that the change in the y-coordinates goes in the numerator and the change in x-coordinates goes in the denominator.

Lesson 1

PROFESSIONAL DEVELOPMENT Learning Progressions

A1_MNLESE368170_U3M06L1 240

3/21/14 3:25 AM

In this lesson, students build on their understanding of linear functions. They focus on the relationships between linear equations and their graphs, including: • The slope-intercept form of a linear equation is y = mx + b, where m represents the slope, and b represents the y-intercept. • A linear function can be graphed by plotting the y-intercept and using the slope to find other points that lie on the line. • The slope-intercept form of a linear equation can be used to write functions that model real-world situations. In future lessons, students compare functions represented in different forms.

Slope-Intercept Form 240



EXPLAIN 2

Substitute (0, 5) for (x 1, y 1) and

INTEGRATE MATHEMATICAL PRACTICES Focus on Reasoning MP.2 Explain to students that one or both

Step 3: Substitute 4 for m and 5 b in the equation y = mx + b. The equation of the line is

13 - 5 8 m=_=_= 4 2 2-0

2

y = mx + b

y = 4x + 5

( 2 )+b

13 = 4

for

13 = 8 + b

.

13 - 8 = 8 + b - 8 5 =b

Write the equation of each line in slope-intercept form. Slope is −1, and (3, 2) is on the line.

4.

2 = -1(3) + b; 5 = b

QUESTIONING STRATEGIES

The line passes through (1, 4) and (3, 18).

18 - 4 14 = 7 m=_=_ 3-1 2 4 = 7(1) + b; -3 = b

The equation of the line is y = -x + 5.

How does the value of b indicate whether the graph is above or below the origin where it intersects the y-axis? If b is positive, the y-intercept is positive and the graph intersects the y-axis above the origin. If b is negative, the y-intercept is negative and the graph intersects the y-axis below the origin.

-2

2

Your Turn

3.

1 x + 2 in two over run. Graph a line such as y = -_ 2 -1 _ and once using a slope ways, once using a slope of 2 1 _ , to show that both result in the same line. of

( 2 , 13 ) for (x , y ).

Step 2: Substitute the slope and x- and y-coordinates of either of the points in the equation y = mx + b.

intercepts are often used to calculate the slope of a linear equation because they are easy to determine. However, any two points that satisfy the given equation can be used to determine the slope.

The equation of the line is y = 7x - 3.

Explain 2

Graphing from Slope-Intercept Form

Writing an equation in slope-intercept form can make it easier to graph the equation. Example 2 © Houghton Mifflin Harcourt Publishing Company

INTEGRATE MATHEMATICAL PRACTICES Focus on Math Connections MP.1 Remind students that slope is the ratio of rise

y2 - y1 m=_ x2 - x1

Step 1: Use the points to find the slope.

Graphing from Slope-Intercept Form

What is the advantage of graphing from slope-intercept form? The intercept is one point on the line and a second point can be found easily by using the slope.

The line passes through (0, 5) and (2, 13).



Write each equation in slope-intercept form. Then graph the line.

y = 5x - 4 The equation y = 5x - 4 is already in slope-intercept form. 5 Slope: m = 5 = _ 1 y-intercept: b = -4 Step 1: Plot (0, -4)

Step 2: Count 5 units up and 1 unit to the right and plot another point. Step 3: Draw a line through the points.

Module 6

241

4

y

2 x -4

-2

0

2

4

-2 -4

Lesson 1

COLLABORATIVE LEARNING A1_MNLESE368170_U3M06L1.indd 241

Peer-to-Peer Activity Group students in pairs. Have each student write slope-intercept equations for four lines: one whose slope is a positive integer, one whose slope is a negative integer, and one whose slope is a fraction. Then have partners trade equations. Partners should first check that the three conditions are met, then graph the lines.

241

Lesson 6.1

3/27/14 9:36 AM



2x + 6y = 6 1 Slope: -_ 3

2x + 6y - 2x = 6 - 2x 6y = -2x + 6

x -4

y-intercept: 1

-2

0

2

Determining Solutions of Equations in Two Variables

4

-2 -4

Step 2: Graph the line.

( 0 , 1 ). Move

EXPLAIN 3

2

_1 y = -3 x + 1

Plot

y

4

Step 1: Write the equation in slope-intercept form by solving for y.

1

3

unit down and

QUESTIONING STRATEGY

units to the

For a real-world problem described by a graph of a linear function in which the value of y indicates the solution for a given value of x, what do you need to do to solve the problem? Apply the units from the graph to the solution. For example if x is time in hours and y is cost in dollars, then the solution is y dollars for a time of x hours.

right to plot a second point. Draw a line through the points. Your Turn

Write each equation in slope-intercept form. Then graph the line. 5.

2x + y = 4

y = -2x + 4 4

6.

2 x+2 y = -_ 3

2x + 3y = 6

y

4

2

y

2 x

-4

-2

0

2

x -4

4

-2 -4

Explain 3

-2

0

2

4

-4

Determining Solutions of Equations in Two Variables



Identify the slope and y-intercept of the graph that represents each linear situation and interpret what they mean. Then write an equation in slope-intercept form and use it to solve the problem.

For one taxi company, the cost y in dollars of a taxi ride is a linear function of the distance x in miles traveled. The initial charge is $2.50, and the charge per mile is $0.35. Find the cost of riding a distance of 10 miles. The rate of change is $0.35 per mile, so the slope, m, is 0.35. The initial cost is the cost to travel 0 miles, $2.50, so the y-intercept, b, is 2.50. Then an equation is y = 0.35x + 2.50.

quantities in a real-world problem, it is usually the independent variable. © Houghton Mifflin Harcourt Publishing Company

Given a real-world linear situation described by a table, a graph, or a verbal description, you can write an equation in slope-intercept form. You can use that equation to solve problems. Example 3

INTEGRATE MATHEMATICAL PRACTICES Focus on Modeling MP.4 Remind students that when time is one of the

-2

AVOID COMMON ERRORS Some students may think that the coefficient of x is the slope of the line of the equation regardless of the form of the equation. Remind them that if the equation is not in the form y = mx + b, the coefficient of x may not be the slope.

y = 0.35x + 2.50

= 0.35(10) + 2.50 =6 (6, 10) is a solution of the equation, and the cost of riding a distance of 10 miles is $6.

To find the cost of riding 10 miles, substitute 10 for x.

Module 6

242

Lesson 1

DIFFERENTIATE INSTRUCTION A1_MNLESE368170_U3M06L1.indd 242

Communicating Math

3/27/14 9:52 AM

Have students list the steps for writing a linear function from two given points. Sample steps are shown. 1. Use the slope formula to find the slope m. 2. Substitute m and the coordinates of one point into f(x) = mx + b. 3. Solve for the y-intercept b. 4. Substitute m and b into f(x) = mx + b.

Slope-Intercept Form 242

B

ELABORATE

A chairlift descends from a mountain top to pick up skiers at the bottom. The height in feet of the chairlift is a linear function of the time in minutes since it begins descending as shown in the graph. Find the height of the chairlift 4 minutes after it begins descending. Height of a Chairlift

QUESTIONING STRATEGIES Height (ft)

How would you graph the equation c = 35t + 50? The equation is in slope-intercept form. 35 is the slope and 50 is the y-intercept. Plot the point that corresponds to the y-intercept (0, 50). Then use the slope to locate a second point on the line. Draw a line through the two points.

0

y (0, 5400)

(4, 2400)

x 1 2 3 4 5 6 7 8 9 Time (min)

The graph contains the points (0, 5400 ) and ( 4 , 2400).

SUMMARIZE THE LESSON

2400 - 5400 The slope is __ = -500 . 4 -0 It represents the rate at which the chairlift descends . The graph passes through the point (0, 5400 ), so the y-intercept is 5400 . It represents the height of the chairlift 0

minutes after it begins descending.

Let x be the time in seconds after the chairlift begins to descend. Let y be the height of the chairlift in feet. The equation is y = -500x + 5400 .

© Houghton Mifflin Harcourt Publishing Company • Image Credits: ©Moodboard/Corbis

How do you write an equation of a line in slope-intercept form when given the slope and y-intercept or when given the slope and a point on the line? Using the form y = mx + b, substitute slope for m and the y-intercept for b. If you are given the slope and a point on the line, substitute the slope into y = mx + b, substitute the coordinates of the point for x and y, and solve for b.

5400 4800 4200 3600 3000 2400 1800 1200 600

To find the height after 4 minutes, substitute 4 for x and simplify.

y =

-500

( 4 ) + 5400

= -2000 + 5400. =

3400

(4, 3400) is a solution of the equation, and the height of the chairlift 4 minutes after it begins descending is

3400

feet.

Reflect

7.

In the example involving the taxi, how would the equation change if the cost per mile increased or decreased? How would this affect the graph? Increasing the cost per mile would increase the value of m and make the graph steeper.

Decreasing the cost per mile would decrease the value of m and make the graph less steep. Module 6

243

Lesson 1

LANGUAGE SUPPORT A1_MNLESE368170_U3M06L1 243

Connect Vocabulary Caution students that a figure called a graph of a line should not be confused with a line graph. A line graph is a graph that uses line segments to connect data points. A graph of a line is a graph of a linear equation.

243

Lesson 6.1

3/21/14 3:25 AM

Identify the slope and y-intercept of the graph that represents each linear situation and interpret what they mean. Then write an equation in slope-intercept form and use it to solve the problem.

EVALUATE

Your Turn

8.

A local club charges an initial membership fee as well as a monthly cost. The cost C in dollars is a linear function of the number of months of membership. Find the cost of the membership after 4 months. - 100 - 277 177 _______ _______ m = 277 = 59; and 454 = ___ = 59, 3-0 6-3 3

Membership Cost Time (months) 0 3 6

Cost ($) 100 277 454

so the rate of change in the cost is $59 per month. The initial cost is $100, so the y-intercept, b, is 100. The equation is y = 59x + 100. f(4) = 59(4) + 100 = 336. So, (4, 336) is a solution.

ASSIGNMENT GUIDE

Elaborate 9.

What are some advantages to using slope-intercept form? When graphing, it’s easy to recognize the slope and y-intercept. It’s also easy to find

y-values for corresponding x-values. 10. What are some disadvantages of slope-intercept form? The x-intercept may not be easily visible, and if a y-value is given, the x-value may not be

easily obtained. 11. Essential Question Check-In When given a real-world situation that can be described by a linear equation, how can you identify the slope and y-intercept of the graph of the equation? To find the slope, identify the rate of change for the situation. To find the y-intercept,

identify the initial value for the situation, that is, the value of the dependent variable

1.

• Online Homework • Hints and Help • Extra Practice

John gets a new job and receives a $500 signing bonus. After that, he makes $200 a day.

The rate of change is $200 per day, so the slope is 200. The value of y

© Houghton Mifflin Harcourt Publishing Company

For each situation, determine the slope and y-intercept of the graph of the equation that describes the situation.

Practice

Explore Graphing Lines Given Slope and y-Intercept

Exercises 1–4

Example 1 Creating Linear Equations in Slope-Intercept Form

Exercises 5–14, 24

Example 2 Graphing from Slope-Intercept Form

Exercises 15–22, 26

Example 3 Determining Solutions of Equations in Two Variables

Exercise 23, 27

INTEGRATE MATHEMATICAL PRACTICES Focus on Reasoning MP.2 Remind students that they can quickly check

when the value of the independent value is 0.

Evaluate: Homework and Practice

Concepts and Skills

that their graphs are reasonable by looking at the slope. Lines with positive slopes rise from left to right, and lines with negative slopes fall from left to right.

when x is 0 (when John has worked 0 days) is $500, so the y-intercept is 500.

AVOID COMMON ERRORS

Module 6

Lesson 1

244

Exercise

A1_MNLESE368170_U3M06L1 244

Depth of Knowledge (D.O.K.)

Encourage students to use a third point to check a graphed line. They can either choose a point from the graph and check it in the equation, or use the equation to generate a point and check that it is on the graph.

COMMON CORE

Mathematical Practices

1–14

1 Recall of Information

MP.6 Precision

15–22

2 Skills/Concepts

MP.6 Precision

23–24

2 Skills/Concepts

MP.4 Modeling

25–26

3 Strategic Thinking

MP.2 Reasoning

27

3 Strategic Thinking

MP.3 Logic

3/21/14 3:25 AM

Slope-Intercept Form 244

2.

KINESTHETIC EXPERIENCE

Jennifer is 20 miles north of her house, and she is driving north on the highway at a rate of 55 miles per hour.

The rate of change is 55 miles per hour, so the slope is 55. The value of

Use masking tape to outline a coordinate plane on a floor of square tiles. Then give a pair of students a length of rope. Announce an equation in slope-intercept form, and have the two students move around on the plane so that when they hold the rope taut, it represents the line described by the given equation, with each of them as two points on the line. A third student can check that these two points satisfy the equation.

y when x is 0 (when Jennifer has driven 0 miles) is 20, so the y-intercept is 20. Sketch a graph that represents the situation. 3.

Morwenna rents a truck. She pays $20 plus $0.25 per mile.

4.

Cost of a Rental Truck

Value of Investment

y

y

28 Amount ($)

620

Cost ($)

26

AVOID COMMON ERRORS

24 22

Students may not believe that they have enough information to find the slope of a line. Remind students that, if you know the equation that describes a line, you can find its slope by using any two ordered-pair solutions.

540 500 x 0

x 0

4

8

12

16

1 2 3 4 5 6 7 8 9 Time (months)

20

Distance (mi)

Write the equation of each line in slope-intercept form. Slope is 3, and (1, 5) is on the line.

6.

The equation is y = -2x + 13.

© Houghton Mifflin Harcourt Publishing Company

The equation is y = 3x + 2.

7.

Slope is _14 , and (4, 2) is on the line.

8.

1 2 = _(4) + b, so 1 = b. 4 1 The equation is y = _x + 1.

The equation is y = 5x - 4.

Slope is -_23 , and (-6, -5) is on the line.

10. Slope is -_12 , and (-3, 2) is on the line.

_2 (-6) + b, so -9 = b. 3 2 x - 9. The equation is y = -_

_

3

A1_MNLESE368170_U3M06L1 245

_ _

1 2 = - 1 (-3) + b, so = b. 2 2 The equation is y = - 1 x + 1 . 2 2

-5 = -

Module 6

Slope is 5, and (2, 6) is on the line.

6 = 5(2) + b, so -4 = b.

4

9.

Slope is -2, and (5, 3) is on the line.

3 = -2(5) + b, so 13 = b.

5 = 3(1) + b, so 2 = b.

Lesson 6.1

580

20

5.

245

An investor invests $500 in a certain stock. After the first six months, the value of the stock has increased at a rate of $20 per month.

245

_

Lesson 1

3/21/14 3:25 AM

12. Passes through (-6, 10) and (-3, -2)

11. Passes through (5, 7) and (3, 1)

m=

6 7-1 _ =_=3 5 -3

m=

2

1 = 3(3) + b, so -8 = b.

10 - (-2) 12 _ = _ = -4 -6 - (-3)

INTEGRATE TECHNOLOGY

-3

Encourage students to use the function graphing capability of a graphing calculator to graph the slope-intercept form and check their answers to the problems. Note that the right side of a function in slope-intercept form can be entered, as the calculator provides Y and the equal sign. Students should experiment with the function grapher and window settings as well as consult their calculator manuals to learn more.

-2 = -4(-3) + b, so -14 = b.

The equation is y = 3x - 8.

The equation is y = -4x - 14.

13. Passes through (6, 6) and (-2, 2)

14. Passes through (-1, -5) and (2, 6)

1 6- 2 4 =_ _ =_ 2 8 6 - (-2) 1( ) _ 2= -2 + b, so 3 = b. 2 1 The equation is y = _x + 3.

-5 - 6 11 -11 = _ _ =_ -1 - 2 3 -3 11 ( ) 4 _ _ 6= 2 + b, so - = b. 3 3 11 4. _ The equation is y = x -_

m=

m=

2

3

3

Write each equation in slope-intercept form. Identify the slope and y-intercept. Then graph the line described by the equation. 15. y = 2x + 3

y = 2x + 3; 2; 3 8

16. y = -x + 2 y = -x + 2; -1; 2

y

8

4

AVOID COMMON ERRORS

y

-8

-4

0

4

x -8

8

-4

-4

-8

2x - 4 17. y = _ 3

0

4

8

-4 -8

2 2 x - 4; - _ y = __ ; -4 3 3

8

1x - 1 1 1 ; -1 18. y = -_ y = - _ x - 1; - __ 2 2 2

y

8

4

y

4

-4

0

4

x -8

8

-4

-4

-8

8

4

8

-4 -8

y = 2x + 5; 2; 5

19. -4x + 2y = 10

0

1 1; 2 y= _ x + 2; __ 2 2

20. 3x - 6y = -12

y

8

4

y

4 x

-8

-4

0

4

x -8

8

-4 -8

Module 6

A1_MNLESE368170_U3M06L1 246

© Houghton Mifflin Harcourt Publishing Company

x -8

Students may have difficulty graphing a function that has a fractional rate of change. Remind them that the fraction can be looked at as rise over run, so the rise and run can be used to move from one point to a second point. That is, if the rate of change is expressed as a fraction, use the numerator to move the appropriate number of units up or down (the rise) and use the denominator to move the appropriate number of units rightward (the run) to plot a second point given the first point.

4 x

-4

0

4

8

-4 -8

246

Lesson 1

3/21/14 3:25 AM

Slope-Intercept Form 246

21. -5x - 2y = 8

COLLABORATIVE LEARNING

8

Have students work in groups of three. Give students the following prompt: “Connor is on a 2-day hike. He hiked 10 miles on Day 1. After 4 hours on Day 2, he had hiked a total of 16 miles.” Have one student draw a graph representing Connor’s progress on Day 2. Have the second student identify the key features (slope and intercept). Have the third student explain what the slope and intercept mean in terms of Connor’s hike. The y-intercept (10) shows how far he hiked on Day 1. The slope (1.5) shows that he hiked an average of 1.5 mi/h on Day 2.

-8

y

8 4

-4

0

4

x -8

8

-4

-4

-8

0

4

8

-4 -8

23. Sports A figure skating school offers introductory lessons at $25 per session. There is also a registration fee of $30. Write a linear equation in slope-intercept form that represents the situation. You want to take at least 6 lessons. Can you pay for those lessons using a $200 gift certificate? If so, how much money, if any, will be left on the gift certificate? If not, explain why not.

© Houghton Mifflin Harcourt Publishing Company • Image Credits: ©YinYang/ Getty Images

24. Represent Real World Problems Lorena and Benita are saving money. They began on the same day. Lorena started with $40. Each week she adds $8. The graph describes Benita’s savings plan. Which girl will have more money in 6 weeks? How much more will she have? Explain your reasoning.

Lorena; $8; the equation for Lorena is y = 40 + 8x, so in 6 weeks, she will save $88. The equation for Benita is y = 50 + 5x, so in 6 weeks, she will save $80.

Amount saved ($)

The cost per lesson is $25, so the slope of the equation that represents the situation is 25. The initial cost of the lessons (that is, before any lessons are paid for), is $30. So the y-intercept is 30. An equation is y = 25x + 30. The cost for 6 lessons is 25(6) + 30 = 180. A $200 gift certificate would pay for the lessons, and $20 would be left.

90 80 70 60 50 40 30 20 10 0

y

x 1 2 3 4 5 6 7 8 9 Time (weeks)

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y

x

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3 3 y = - _ x - 3; - _ ; -3 4 4

22. 3x + 4y = -12

4

JOURNAL Have students show different representations of a linear function: the linear equation, the slope-intercept form of the equation, the graph, and a description of the relationships.

5 5 y = - _ x - 4; - __ ; -4 2 2

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QUESTIONING STRATEGIES

H.O.T. Focus on Higher Order Thinking

25. Analyze Relationships Julio and Jake start their reading assignments the same day. Jake is reading a 168-page book at a rate of 24 pages per day. Julio’s book is 180 pages long and his reading rate is 1__14 times Jake’s rate. After 5 days, who will have more pages left to read? How many more? Explain your reasoning.

How do the graphs of Year 1 and Year 2 compare? They are parallel lines having the same slope, but different y-intercepts.

Jake will have 48 more pages left to read. For Jake, the number of pages to read after 0 days is 168, and the rate of change is -24, so the equation is y = -24x + 168. Because -24(5) + 168 = 48, Jake will have 48 pages left to read after 5 days. Julio’s 5( ) 24 , so the equation for Julio is y = -30x + 180. Then after 5 days, Julio pace is _ 4 will have -30(5) + 180 = 30 pages left to read, and 48 - 30 = 18.

How do the y-values compare for any whole-number x-value? What does this indicate about the costs? y is always 40 more for Year 2 than Year 1; Year 2 costs $40 more for any number of months.

So, Jake has 48 - 30 = 18 more pages to read.

26. Explain the Error John has $2 in his bank account when he gets a job. He begins making $107 dollars a day. A student found that the equation that represents this situation is y = 2x + 107. What is wrong with the student’s equation? Describe and correct the student’s error.

INTEGRATE MATHEMATICAL PRACTICES Focus on Reasoning MP.2 Students can check each year‘s equation for

The student switched the slope and the y-intercept. The slope should be 107, and the y-intercept should be 2. So the equation is y = 107x + 2. 27. Justify Reasoning Is it possible to write the equation of every line in slopeintercept form? Explain your reasoning.

No; it is not possible to write the equation of a vertical line in slope-intercept form. The equation of a vertical line has form x = a, where a is a real number. The slope of a vertical line is undefined.

correctness by substituting the values of two ordered pairs from the graph of each line into its equation and verifying that both solutions make the equation true. Remind students that checking their equations requires 2 points to define a line, so at least 2 points must be checked.

Lesson Performance Task Gym Memberships

b. What are the values that represent the sign-up fee and the membership cost? How did the values change between the years?

a. The equation for Year 1 is y = 20x + 20, and the equation for Year 2 is y = 20x + 60.

Cost (Dollars)

a. Write an equation in slope-intercept form for each of the two lines in the graph.

180 160 140 120 100 80 60 40 20 0

y Year 2

Year 1 x 1 2 3 4 5 6 7 8 9 10 Time (Months)

b. The y-intercepts of the two graphs are 20 and 60, so these both represent the sign-up fees. In year 1, the graph increases by $20 every month. Since this represents a rate, the slope of the line in year 1 is 20. In year 2, the graph again increases by $20 every month. So the slopes are equal in both years. This means that the monthly membership fees did not change. However, the sign-up fee increased by $40 between years 1 and 2.

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The graph shows the cost of a gym membership in each of two years. What are the values that represent the sign-up fee and the membership cost? How did the values change between the years?

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Have students research the cost of joining two gyms. Have students write an equation to represent the cost of each gym. Then have students graph their equations on the same coordinate grid. Students will find that some gyms have a higher initial fee, but lower monthly rates than others. Caution students to note whether the fees to attend are weekly or monthly.

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Scoring Rubric 2 points: Student correctly solves the problem and explains his/her reasoning. 1 point: Student shows good understanding of the problem but does not fully solve or explain his/her reasoning. 0 points: Student does not demonstrate understanding of the problem.

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