Download Slope-Intercept Form

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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Essential

Lesson 1
239
Module 6
L1 239
0_U3M06
SE36817
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)
A1_MNLESE368170_U3M06L1 247
Lesson 6.1
y
x
Module 6
247
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
247
Lesson 1
3/21/14 3:25 AM
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.
Module 6
248
© Houghton Mifflin Harcourt Publishing Company
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?
Lesson 1
EXTENSION ACTIVITY
A1_MNLESE368170_U3M06L1 248
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.
3/21/14 3:25 AM
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.
Slope-Intercept Form 248