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LESSON
6.1
Name
Class
Date
Slope-Intercept Form
6.1
Texas Math Standards
Essential Question: How can you represent a linear function in a way that reveals its slope
and y-intercept?
The student is expected to:
A1.2.B …write linear equations in two variables in various forms, including y = mx + b…
Also A1.2.C, A1.3.C
A1.2.B
Write linear equations in two variables in various forms, including
y = mx + b..., given one point and the slope and given two points.
Also A1.2.C, A1.3.C
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.
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.
A1.1.D
Communicate mathematical ideas, reasoning, and their implications
using multiple representations, including symbols, diagrams, graphs, and
language as appropriate.
time is –$50 per month.
2.C.3, 2.C.4, 2.I.3, 2.I.4, 3.D, 4.G
Explain to a partner how to write a linear function in
slope-intercept form.
The y-intercept of the graph of the equation that represents
the situation is 500 .
The slope is –50 .
Use the y-intercept to plot a point on the graph of the equation. The y-intercept is 500 ,
so plot the point (0, 500) .
Amount Andrew Owes
Amount ($)
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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.
When x = 0, y = $500 .
The rate of change in the amount Andrew owes over
Language Objective
Essential Question: How can you
represent a linear function in a way that
reveals its slope and y-intercept?
500
450
400
350
300
250
200
150
100
50
0
y
x
1 2 3 4 5 6 7 8 9
Time (Months)
PREVIEW: LESSON
PERFORMANCE TASK
View the Engage section online. Discuss the photo
and how a gym membership may require a one-time
sign-up fee as well as regular monthly fees. Also
discuss how a graph of this type of data might look.
Then preview the Lesson Performance Task.
Module 6
Lesson 1
247
Date
Class
Name
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Slope-Inter
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6.1
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Time (Month
Lesson 1
247
Module 6
247 Lesson 6.1
Resource
Locker
Graphing Lines Given Slope and y-intercept
Explore
Mathematical Processes
ENGAGE
Slope-Intercept Form
Using the definition of slope, plot a second point.
–50
Change in y
Slope = __ = _ = –50 .
1
Change in x
Start at the point you plotted. Count 50 units
down and 1 unit right and plot another point.
Amount ($)
500
450
400
350
300
250
200
150
100
50
0
Draw a line through the points you
plotted.
y
EXPLORE
Amount Andrew Owes
Amount ($)
Amount Andrew Owes
500
450
400
350
300
250
200
150
100
50
0
Graphing Lines Given Slope and
y-Intercept
y
INTEGRATE TECHNOLOGY
Students have the option of completing the activity
either in the book or online.
x
1 2 3 4 5 6 7 8 9
Time (Months)
CONNECT VOCABULARY
x
1 2 3 4 5 6 7 8 9
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.
Time (Months)
Reflect
1.
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 unit to the right. Then repeat this process beginning at the new point.
2.
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
to be repaid is $0.
Explain 1
© Houghton Mifflin Harcourt Publishing Company
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 .
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
mx = y - b
mx + b = y
Multiply both sides by x(x ≠ 0).
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.
Module 6
EXPLAIN 1
Creating Linear Equations in Slope-Intercept Form
248
Lesson 1
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.
PROFESSIONAL DEVELOPMENT
Learning Progressions
In this lesson, students build on their understanding of linear functions. They
focus on the relationships between linear equations and their graphs, including:
r The slope-intercept form of a linear equation is y = mx + b, where m
represents the slope, and b represents the y-intercept.
r A linear function can be graphed by plotting the y-intercept and using the
slope to find other points that lie on the line.
r 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
248
Example 1
Write the equation of each line in slope-intercept form.
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
Write the slope–intercept form.
Substitute 3 for m, 2 for x, and 5 for y.
Multiply.
Subtract 6 from both sides.
Simplify.
Step 2: Write the equation.
y = mx + b
y = 3x + (-1)
y = 3x - 1
Write the slope–intercept form.
Substitute 3 for m and -1 for b.
The line passes through (0, 5) and (2, 13).
Step 1: Use the points to find the slope.
y2 - y1
The slope formula is m = _.
x2 - x1
Substitute (0, 5) for (x 1, y 1) and
( 2 , 13 ) for (x , y ).
2
2
© Houghton Mifflin Harcourt Publishing Company
8
13 - 5
m=_=_= 4
2
2-0
( 2 )+b
Step 2: Substitute the slope and x- and y-coordinates
of either of the points in the equation y = mx + b.
13 = 4
4
Substitute
for m and the x- and y-coordinates
of the point (2, 13) for x and y.
13 = 8 + b
4 for m and 5
Step 3: Substitute
b in the equation y = mx + b.
The equation of the line is
y = 4x + 5
13 - 8 = 8 + b - 8
for
5 =b
.
Module 6
249
Lesson 1
COLLABORATIVE LEARNING
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.
249 Lesson 6.1
Reflect
3.
EXPLAIN 2
Discussion How would the equation change if (0, 5) were used for (x 2, y 2) and (2, 13) were used
for (x 1, y 1) in the slope formula? Explain your reasoning.
The equation would not change at all. It doesn’t make a difference which point
Graphing from Slope-Intercept Form
-5
- 13
_____
is used for (x 1, y 1) and which point is used for (x 2, y 2) because 13
= 5_____
.
2-0
0-2
INTEGRATE MATHEMATICAL
PROCESSES
Focus on Reasoning
Your Turn
Write the equation of each line in slope-intercept form.
4.
Slope is −1, and (3, 2) is on the line.
5.
The line passes through (1, 4) and (3, 18).
18 - 4 _
= 14 = 7
3-1
2
y = mx + b
y = mx + b
m=
2 = -1(3) + b
5= b
Explain to students that one or both 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.
4 = 7(1) + b
The equation of the line is y = -x + 5.
-3 = b
The equation of the line is y = 7x
7 - 3.
Explain 2
Graphing from Slope-Intercept Form
Writing an equation in slope-intercept form can often make it easier to graph the equation.
Example 2
QUESTIONING STRATEGIES
Write each equation in slope-intercept form. Then graph the line described by the equation.
y = 5x - 4
4
The equation y = 5x - 4 is already in slope-intercept form.
5
Slope: m = 5 = _
1
y-intercept: b = -4
2
x
-4
4
0
-2
2
-2
Step 2: Count 5 units up and 1 unit to the right and plot another point.
-4
4
2x + 6y
6 =6
4
Step 1: Write the equation in slope-intercept form by solving for y.
2
1
Slope: -_
3
2x + 6y
6 - 2x = 6 - 2x
6 = -2x + 6
6y
y-intercept: 1
y
x
-4
4
0
-2
2
2
4
-2
_1
y = -3 x + 1
-4
Step 2: Graph the line.
r Plot
( 0 , 1 ).
r Move
1
unit down and
3
units to the right to plot a second point.
r Draw a line through the points.
Module 6
250
Lesson 1
© Houghton Mifflin Harcourt Publishing Company
Step 1: Plot (0, -4).
2
Step 3: Draw a line through the points.
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.
y
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.
INTEGRATE MATHEMATICAL
PROCESSES
Focus on Math Connections
Remind students that slope is the ratio of rise over
1
run. Graph a line such as y = -_
x + 2 in two ways,
2
-1
_
once using a slope of
and once using a slope of
2
1
_
, to show that both result in the same line.
-2
DIFFERENTIATE INSTRUCTION
Communicating Math
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
250
Your Turn
EXPLAIN 3
Write each equation in slope-intercept form. Then graph the line described by
the equation.
6.
Determining Solutions of Equations
in Two Variables
2x + y = 4
y = -2x + 4
4
7.
2x + 3y = 6
2
x+2
y = -_
3
y
4
2
2
QUESTIONING STRATEGY
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.
Explain 3
0
-2
2
-4
4
0
-2
-2
-2
-4
-4
2
4
Determining Solutions of Equations in Two Variables
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
PROCESSES
Focus on Modeling
Remind students that when time is one of the
quantities in a real-world problem, it is usually the
independent variable.
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.
© Houghton Mifflin Harcourt Publishing Company
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.
x
x
-4
AVOID COMMON ERRORS
y
Then an equation is y = 0.35x + 2.50.
To find the cost of riding a distance of 10 miles, substitute 10 for x and simplify.
y = 0.35x + 2.50
= 0.35(10) + 2.50
= 3.5 + 2.5
=6
(6, 10) is a solution of the equation, and the cost of riding a distance of 10 miles is $6.
Module 6
251
Lesson 1
LANGUAGE SUPPORT
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.
251 Lesson 6.1
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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 2 minutes after it begins descending.
Height (ft)
Height of a Chairlift
y
5400 (0, 5400)
4800
4200
(2, 3900)
3600
3000
(4, 2400)
2400
1800
1200
600
0
x
1 2 3 4 5 6 7 8 9
Time (min)
4 , 2400).
The graph contains the points (0, 5400 ) and (
2400 - 5400
The slope is __ = -750 .
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 minutes after the chairlift begins to descend.
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Let y be the height of the chairlift in feet.
The equation is y = -750x + 5400 .
To find the height after 2 minutes, substitute 2 for x and simplify.
y =
-750
( 2 ) + 5400
= -1500 + 5400
=
3900
(2, 3900) is a solution of the equation, and the height of the chairlift 2 minutes after it begins
descending is
3900
feet.
Reflect
8.
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
252
Lesson 1
Slope-Intercept Form
252
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.
ELABORATE
Your Turn
QUESTIONING STRATEGIES
9.
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.
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.
Membership Cost
Time (months)
0
3
6
Cost ($)
100
277
454
277
- 100
- 277
177
177
_______
_______
= ___
= 59 and 454
= ___
= 59, so the rate of change in the cost is $59 per
3-0
3
6-3
3
month. The slope, m, is 59.
The initial cost is the cost for 0 months, $100, so the y-intercept, b, is 100.
Let x be the number of months and y be the cost in dollars. The equation is y = 59x + 100.
When x = 4, y = 59(4) + 100 = 336. So, (4, 336) is a solution of the equation, and the
membership costs $336 for 4 months.
SUMMARIZE THE LESSON
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.
Elaborate
10. 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.
11. 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
© Houghton Mifflin Harcourt Publishing Company
easily obtained.
12. 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
when the value of the independent value is 0.
Module 6
253 Lesson 6.1
253
Lesson 1