LESSON
6.2
Name
Point-Slope Form
Class
6.2
Point-Slope Form
Essential Question: How can you represent a linear function in a way that reveals its slope
and a point on its graph?
Texas Math Standards
The student is expected to:
A1.2.B …write linear equations in two variables in various forms, including…
y – y 1 = m(x – x 1)… Also A1.2.C
A1.2.B
Explore
Write linear equations in two variables in various forms, including
y - y 1 = m(x - x 1)..., given one point and the slope and given two
points. Also A1.2.C

A1.1.F
Analyze mathematical relationships to connect and communicate
mathematical ideas.
y- 1
4 =_
x- 2
2.C.3, 2.C.4, 2.I.3, 2.I.4, 3.D, 4.G

Use the Multiplication Property of Equality to get rid of the fraction.
Explain to a partner how to write a linear function in point-slope form.
4
ENGAGE
PREVIEW: LESSON
PERFORMANCE TASK
View the Engage section online. Discuss the photo
and how the concept of slope applies to
snowboarding and can be used to predict the
snowboarder’s height on the slope at any one specific
time. Then preview the Lesson Performance Task.
Deriving Point-Slope Form
A line has a slope m of 4, and the point (2, 1) is on the line. Let (x, y) be any other point on
the line. Substitute the information you have in the slope formula.
y2 - y1
m=_
x2 - x1
Language Objective

(x - 2 ) =
Simplify.
4
© Houghton Mifflin Harcourt Publishing Company
You can determine the slope m of the graph of
the function and the coordinates (x 1, y 1) of a point
on the graph, then write the equation y - y 1 = m
(x - y 1), which is called the point-slope form of the
equation.
( )(
y- 1
_
x- 2
x-
)
2
(x - 2 ) = (y - 1 )
Reflect
1.
Discussion The equation that you derived is written in a form called point-slope form. The equation
y = 2x + 1 is in slope-intercept form. How can you rewrite it in point-slope form?
The slope is 2. The y-intercept is 1, so the point (0, 1) is on the line. You can use the
method of the Explore, that is, substitute the known values in the slope formula, write the
equation without a fraction, and simplify the result. y - 1 = 2 (x - 0)
Explain 1
Creating Linear Equations Given Slope and a Point
Point-Slope Form
The line with slope m that contains the point (x 1, y 1) can be described by the equation y - y 1 = m(x - x 1).
Module 6
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Lesson 2
259
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Lesson 2
259
Module 6
Lesson 6.2
L2.indd
4_U3M06
SE34673
A1_MTXE
259
Resource
Locker
Suppose you know the slope of a line and the coordinates of one point on the line. How can you write an equation of
the line?
Mathematical Processes
Essential Question: How can you
represent a linear function in a way
that reveals its slope and a point on
its graph?
Date
259
2/22/14
4:55 AM
2/22/14 4:55 AM
Example 1

Write an equation in point-slope form for each line.
Slope is 3.5, and (-3, 2) is on the line.
y - y 1 = m(x - x 1)
y - 2 = 3.5(x - (-3))
y - 2 = 3.5(x + 3)

EXPLORE
Slope is 0, and (-2, -1) is on the line.
Point-slope form
y - y 1 = m(x - x 1)
Point-slope form
Substitute.
y - (-1) = 0
Substitute.
Simplify.
y+ 1 = 0
(x - (-2) )
Deriving Point-Slope Form
Simplify.
INTEGRATE TECHNOLOGY
Students have the option of completing the activity
either in the book or online.
Reflect
2.
Communicate Mathematical Ideas Suppose that you are given that the slope of a line is 0. What is the
only additional information you need to write an equation of the line? Explain.
The y-coordinate y 1 of any point on the line. A line with slope 0 is horizontal. Every point
has the same y-coordinate, and the equation of the line is y = -y 1.
CONNECT VOCABULARY
Point out that the names slope-intercept and
point-slope tell students what types of information
they will use to express a linear relationship.
Your Turn
Write an equation in point-slope form for each line.
3.
Slope is 6, and (1, 2) is on the line.
4.
y - y 1 = m(x - x 1)
1 x+3
y-1=_
)
(
3
y - y 1 = m(x - x 1)
y - 2 = 6(x - 1)
Explain 2
Slope is _13 , and (-3, 1) is on the line.
EXPLAIN 1
Creating Linear Models Given Slope and a Point
Creating Linear Equations Given
Slope and a Point
You can write an equation in point-slope form to describe a real-world linear situation. Then you can use that
equation to solve a problem.
Example 2
Paul wants to place an ad in a newspaper. The newspaper charges $10 for the first 2 lines of text
and $3 for each additional line of text. Paul’s ad is 8 lines long. How much will the ad cost?
Let x represent the number of lines of text. Let y represent the cost in dollars of the ad.
Because 2 lines of text cost $10, the point (2, 10) is on the line. The rate of change in the
cost is $3 per line, so the slope is 3.
Write an equation in point-slope form.
y - y 1 = m(x - x 1)
Point-slope form
y − 10 = 3(x − 2)
Substitute 3 for m, 2 for x 1, and 10 for y 1.
To find the cost of 8 lines, substitute 8 for x and solve for y.
y - 10 = 3(8 - 2)
Substitute
y − 10 = 18
Simplify.
INTEGRATE MATHEMATICAL
PROCESSES
Focus on Critical Thinking
© Houghton Mifflin Harcourt Publishing Company

Solve the problem using an equation in point-slope form.
Show students that they can graph a line starting at
any point on the line if they know the slope, by
counting vertically and horizontally from that point.
Show how point-slope form comes from the slope
formula, and show how it also simplifies to
slope-intercept form.
y = 28
QUESTIONING STRATEGIES
The cost of 8 lines is $28.
Module 6
260
Lesson 2
PROFESSIONAL DEVELOPMENT
A1_MTXESE346734_U3M06L2 260
Integrate Mathematical Processes
This lesson provides an opportunity to address Mathematical Process
TEKS A1.1.F, which calls for students to “analyze mathematical relationships
to connect and communicate mathematical ideas.” Students learn how point-slope
form is related to slope-intercept form, and decide which form works best in a
situation.
11/5/14 1:27 PM
Could two different equations in point-slope
form represent the same line? Yes; choosing
two different points on a line would result in two
different equations in point-slope form, although
the slope would be the same.
Point-Slope Form
260

EXPLAIN 2
Paul would like to shop for the best price to place the ad. A different newspaper has a base
cost of $15 for 3 lines and $2 for every extra line. How much will an 8-line ad cost in
this paper?
y - y 1 = m(x - x 1)
Creating Linear Models Given Slope
and a Point
(
= 2(
y - 15 = 2 x - 3
y - 15
8 - 3
y - 15 = 10
INTEGRATE MATHEMATICAL
PROCESSES
Focus on Reasoning
)
Point-slope form
)
y = 25
Substitute.
Substitute for x.
Simplify the right side.
Solve for y.
The cost of 8 lines is $ 25 .
Remind students that once they have the point-slope
equation of a line, they can find the value of y if given
a value of x, or vice versa, by substituting and solving
for the other variable.
Reflect
5.
Analyze Relationships Suppose that you find that the cost of an ad with 8 lines in another publication
is $18. How is the ordered pair (8, 18) related to the equation that represents the situation? How is it related
to the graph of the equation?
The ordered pair is a solution of the equation. It represents a point on the graph of
the equation.
QUESTIONING STRATEGIES
EXPLAIN 3
Creating Linear Equations Given Two
Points
Your Turn
6.
Daisy purchases a gym membership. She pays a signup fee and a monthly fee of $11. After 4 months, she
has paid a total of $59. Use a linear equation in point-slope form to find the signup fee.
y − 59 = 11(x − 4)
y − 59 = 11(0 − 4)
y = 15 The signup fee is $15.
Explain 3
© Houghton Mifflin Harcourt Publishing Company
If the independent variable is the number of
items and the dependent variable is the weight
of the items in pounds, what would the equation in
point-slope form represent? y minus the weight in
pounds of a number of items would be equal to the
product of the weight in pounds per item and the
quantity x minus the number of items.
You can use two points on a line to create an equation of the line in point-slope form. There is more than one such
equation.
Example 3

Write an equation in point-slope form for each line.
Let (2, 1) = (x 1, y 1) and let (3, 4) = ( x 2, y 2).
Find the slope of the line by substituting the
given values in the slope formula.
y2 - y1
m=_
x -x
4-1
=_
3-2
=3
QUESTIONING STRATEGIES
Can you write a point-slope equation for a
horizontal line? Explain. Yes; the slope would
be 0, so the equation would be in the form
y - y 1 = 0.
261
Lesson 6.2
You can choose either point and substitute the
coordinates in the point-slope form.
(2, 1) and (3, 4) are on the line.
2
Can you write a point-slope equation for a
vertical line? Explain. No; because a vertical
line has rise, but no run, it has undefined slope.
Creating Linear Equations Given Two Points
Point-slope form
y - 1 = 3(x - 2)
Substitute 3 for m,
2 for x 1, and 1 for y 1.
Or:
1
Module 6
y - y 1 = m(x - x 1)
261
y - y 1 = m(x - x 1)
Point-slope form
y - 4 = 3(x - 3)
Substitute 3 for m,
3 for x 1, and 4 for y 1.
Lesson 2
COLLABORATIVE LEARNING
A1_MTXESE346734_U3M06L2 261
Small Group Activity
Have students work with in groups of three. Have each student write the
coordinates for a point and the slope of a line. Remind students to keep their
values reasonable. Students pass their data to the right. The next student writes the
point-slope form for the line. When students are finished, pass the papers to the
right once more. Now, students match the form to the data; when they are satisfied
it is correct, they graph the line.
11/5/14 1:28 PM

Choose either point and substitute the coordinates in
the point-slope form.
(1, 3) and (2, 3) are on the line.
Let (1, 3) = (x 1, y 1) and let (2, 3) = (x 2, y 2).
y - y 1 = m(x - x 1)
Find the slope of the line by substituting the
given values in the slope formula.
y2 - y1
m=_
x2 - x1
y- 3 = 0
(x - 1 )
AVOIDING COMMON ERRORS
Point-slope form
Students sometimes substitute for the variables in the
opposite order, assuming that x comes first as in the
ordered pair (x, y). Remind them that the value of y is
substituted on the left side of the equation, while the
value of x is substituted on the right side.
Substitute 0 for m, 1
for x 1, and 3 for y 1.
Or:
3 - 3
= __
2 - 1
y - y 2 = m(x - x 2)
y- 3 = 0
= 0
(x - 2 )
Point-slope form
Substitute 0 for m, 2
for x 2, and 3 for y 2.
EXPLAIN 4
Reflect
7.
Given two points on a line, Martin and Minh each found the slope of the line. Then Martin used (x 1, y 1)
and Minh used (x 2, y 2) to write the equation in point-slope form. Each student’s equation was correct.
Explain how they can show both equations are correct.
Martin can show that (x 2 , y 2) is a solution of his equation, and Minh can show that
Creating Linear Models Given Two
Points
(x 1 , y 1) is a solution of hers.
QUESTIONING STRATEGIES
Your Turn
How do you determine the intercepts given
two points? Use the points to find the slope.
Replace y with 0 and solve for x. Solve the
slope-intercept form for y.
Write an equation in point-slope form for each line.
8.
(2, 4) and (3, 1) are on the line.
9.
Explain 4
y - 1 = 0(x -0) or y - 1 = 0(x - 1)
Creating a Linear Model Given Two Points
In a real-world linear situation, you may have information that represents two points on the line. You can write an
equation in point-slope form that represents the situation and use that equation to solve a problem.
Example 4
Solve the problem using an equation in point-slope form.
An animal shelter asks all volunteers to take a training session and
then to volunteer for one shift each week. Each shift is the same
number of hours. The table shows the numbers of hours Joan and her
friend Miguel worked over several weeks. Another friend, Lili, plans
to volunteer for 24 weeks over the next year. How many hours will Lili
volunteer?
Volunteer
Weeks worked
Hours worked
6
15
10
23
Joan
Miguel
Module 6
262
© Houghton Mifflin Harcourt Publishing Company • Image Credits: ©Mila
Supinskaya/Shutterstock
y - 1 = -3(x -3) or y - 4 = -3(x - 2)
(0, 1) and (1, 1) are on the line.
Lesson 2
DIFFERENTIATE INSTRUCTION
A1_MTXESE346734_U3M06L2 262
Auditory Cues
06/11/14 8:28 PM
Students will benefit from saying the point-slope equation aloud when writing it.
They might say, for example:
“y minus y one equals m times the quantity x minus x one,”
“ y minus y sub one equals m times the quantity x minus x sub one,”
“ y minus y one equals the product of m and x minus x one.”
Be sure they state it in a way that emphasizes the distribution of m.
Point-Slope Form
262
Analyze Information
Identify the important information.
6
15
weeks for a total of
hours.
• Joan worked for
10
23
• Miguel worked for
weeks for a total of
hours.
24
• Lili will work for
weeks.
Formulate a Plan
To create the equation, identify the two ordered pairs represented by the situation. Find the
slope of the line that contains the two points. Write the equation in point-slope form. Substitute
the
number of weeks
that Lili works for x to find y, the number of hours that Lili works .
Let x represent the number of weeks worked and y represent the number of hours worked.
6, 15) and (10, 23) are on the line. Substitute the coordinates in the slope formula
The points (
to find the slope.
y2 - y1
m=_
x2 - x1
23 - 15
m = __
10 - 6
m= 2
Next choose one of the points and find an equation of the line in point-slope form.
y - y 1 = m(x - x 1)
y - 15 = 2
Or:
Finally, substitute
© Houghton Mifflin Harcourt Publishing Company
(x - 6 )
y - y 2 = m(x - y 2)
y - 23 = 2
Point-slope form
(x 24
10
Substitute 2 for m, 6 for x 1, and 15 for y 1.
)
Point-slope form
Substitute 2 for m, 10 for x 2, and 23 for y 2.
in the equation to find y.
(x - 6 )
2 ( 24 - 6 )
2 ( 18 )
y - 15 = 2
Substitute 2 for m, 6 for x 1, and 15 for y 1.
y - 15 =
Substitute 24 for x.
y - 15 =
Or:
y = 51
Simplify.
Simplify.
(x - 10 ) Substitute
2 ( 24 - 10 ) Substitute
Simplify.
2 ( 14 )
y - 23 = 2
2 for m, 10 for x 2, and 23 for y 2.
y - 23 =
24 for x.
y - 23 =
y = 51
Lili will work a total of
Simplify.
51
Module 6
hours.
263
Lesson 2
LANGUAGE SUPPORT
A1_MTXESE346734_U3M06L2 263
Connect Vocabulary
Explain to students how some English words are formed from two hyphenated
words, such as real-world, family-size, and in this lesson, point-slope. The hyphen
in these words links the two source words together. Together, point-slope is one
compound word that describes the form of a linear equation. Contrast this with
the slope-intercept form of a line, pointing out that each compound adjective
describes the information in its equation.
263
Lesson 6.2
11/5/14 2:12 PM
Justify and Evaluate
The ordered pair
(
given information.
24 ,
51
) is a solution of both equations obtained using the
(x - 6 )
2 ( 24 - 6 )
y - 15 = 2
51 - 15 =
Substitute 2 for m, 6 for x 1, and 15 for y 1.
Substitute 24 for x and 51 for y.
36 = 36
Or:
(
2 (
Simplify.
y - 23 = 2 x - 10
51 - 23 =
)
24 - 10
28 = 28
)
Substitute 2 for m, 10 for x 2, and 23 for y 2.
Substitute 24 for x and 51 for y.
Simplify.
The answer makes sense because the rate of change in the number of hours is the
slope,
23 + 2
2
(14)
14
. Because Lili will work
hours, or
51
more weeks than Miguel, she will work
hours.
Your Turn
49 - 10
m=_=3
18 - 5
A member would pay $61
for 22 gallons of gas.
y - 10 = 3(x - 5)
y - 10 = 3(22 - 5)
y = 61
50
45
40
35
30
25
20
15
10
5
0
11. A roller skating rink offers a special rate for birthday parties.
On the same day, a party for 10 skaters cost $107 and a party
for 15 skaters cost $137. How much would a party for 12 skaters cost?
137 - 107
m=_=6
15 - 10
A party for 12 skaters would cost $119.
Module 6
A1_MTXESE346734_U3M06L2.indd 264
y
(18, 49)
(5, 10)
x
2 4 6 8 10 12 14 16 18 20
Amount (gal)
© Houghton Mifflin Harcourt Publishing Company
10. A gas station has a customer loyalty program. The graph shows
the amount y dollars that two members paid for x gallons
of gas. Use an equation in point-slope to find the amount a
member would pay for 22 gallons of gas.
Cost ($)
Solve the problem using an equation in point-slope form.
y - 107 = 6(x - 10)
y - 107 = 6(12 - 10)
y = 119
264
Lesson 2
2/16/14 11:26 AM
Point-Slope Form
264
Elaborate
ELABORATE
12. Can you write an equation in point-slope form that passes through any two given points in a
coordinate plane?
No; you can’t write a linear equation given two points with the same x-coordinate. A line
QUESTIONING STRATEGIES
through two such points is vertical and has no slope.
How is the point-slope form related to the
slope formula? The point-slope form is just
the slope formula with the denominator (x-values)
moved to the other side of the equal sign.
13. Compare and contrast the slope-intercept form of a linear equation and the point-slope form.
Possible answers: Both forms of the equation reveal the slope. The point-slope form
explicitly reveals a point (x 1, y 1) on the line. The slope-intercept form reveals a point on
the line, but not explicitly. It is the point (0, b) where the graph intersects the y-axis. Both
can be used fairly easily to graph the function. You can plot (0, b) or (x 1, y 1), and then
use the slope to plot a second point. The slope-intercept form can be used to graph an
INTEGRATE MATHEMATICAL
PROCESSES
Focus on Critical Thinking
equation on a graphing calculator.
14. Essential Question Check-In Given a linear graph, how can you write an equation in point-slope form
of the line?
Find the slope m (if it is defined) by identifying two points on the line and using the
Ask students to describe the graph of a linear
equation with a point-slope form of y - 3 = 3(x - 1)
without simplifying the equation. Students should
recognize that the slope is 3 and the graph goes
through (1, 3), so the graph is the line with slope 3
going through the origin, or y = 3x.
slope formula. Then substitute the slope and the coordinates of one of the points in the
point-slope form y - y 1 = m(x - x 1).
Evaluate: Homework and Practice
1.
Yes; the equation is equivalent to y - (-1) = 7(x - (-2)).
SUMMARIZE THE LESSON
2.
Slope is 1 and (-2, -1) is on the line.
3.
y - y 1 = m(x - x 1)
4.
Slope is 0, and (1, 2) is on the line.
y - 1 = (-2)(x - 1)
5.
(_)
y - 2 = 0(x - 1)
(1, 6) and (2, 3) are on the line.
7.
3−6
_
= −3
Lesson 6.2
(-1) - 1 _
-2
_
=
= -1
2
1 - (-1)
y − y 1 = m(x − x 1)
y − 6 = (−3)(x − 1) or y − 3 = (−3)(x − 2)
A1_MTXESE346734_U3M06L2 265
(-1, 1) and (1, -1) are on the line.
m=
2−1
y − y 1 = m(x − x 1)
Module 6
Slope is __14 , and (1, 2) is on the line.
y - y 1 = m(x - x 1)
1 (
y-2=
x - 1)
4
y - y 1 = m(x - x 1)
6.
Slope is -2, and (1, 1) is on the line.
y - y 1 = m(x - x 1)
y + 1 = 1(x + 2)
m=
265
• Online Homework
• Hints and Help
• Extra Practice
Write an equation in point-slope form for each line.
© Houghton Mifflin Harcourt Publishing Company
How do you write linear equations in
point-slope form if you know a point and the
slope, or if you know two points? The point-slope
form, y - y 1 = m (x - x 1), uses the x- and
y-coordinates of a point and the slope. If you are
given two points, use the slope formula to find the
slope. Substitute the slope and the coordinates of a
given point into the point-slope form.
Is the equation y + 1 = 7(x + 2) in point-slope form? Justify your answer.
y − 1 = (-1)(x + 1) or y + 1 = (-1)(x - 1)
265
Lesson 2
11/5/14 2:15 PM
8.
(7, 7) and (−3, 7) are on the line.
0
7−7
m=_=_=0
9.
(0, 3) and (2, 4) are on the line.
m=
(−3) − 7 −10
y − y 1 = m(x − x 1)
EVALUATE
4−3 _
1
_
=
2
2−0
y − y 1 = m(x − x 1)
1 (
1 (
y−3=
x - 0) or y - 4 =
x - 2)
2
2
y − 7 = 0(x − 7) or y − 7 = 0(x + 3)
(_)
Solve the problem using an equation in point-slope form.
10. An oil tank is being filled at a constant rate. The depth of the oil
is a function of the number of minutes the tank has been filling,
as shown in the table. Find the depth of the oil one-half hour after
filling begins.
(_)
Time (min)
5-3
1
m = _ = _
10 - 0 5
y − y 1 = m(x − x 1)
1 (x − 0) or y − 5 = _
1 (x − 10)
y−3=_
5
5
1 (30 − 0)
y−3=_
5
y−3=6
Depth (ft)
ASSIGNMENT GUIDE
0
3
10
5
Concepts and Skills
Practice
15
6
Explore
Deriving Point-Slope Form
Exercise 1
Example 1
Creating Linear Equations Given
Slope and a Point
Exercises 2–5
Example 2
Creating Linear Models Given Slope
and a Point
Exercises 10–12,
16–20
Example 3
Creating Linear Equations Given Two
Points
Exercises 6–9,
21–22
Example 4
Creating A Linear Model Given Two
Points
Exercises 13–15,
23–24
y=9
One-half hour after filling begins, the depth of the oil is 9 feet.
11. James is participating in a 5-mile walk to raise money for a charity. He has received $200 in fixed pledges
and raises $20 extra for every mile he walks. Use a point-slope equation to find the amount he will raise if
he completes the walk.
y - 200 = 20(5 - 0)
y - 200 = 100
y = 300
If he finishes the race, James will raise $300.
y - 325 = -25(10 - 0)
y - 325 = -250
y = 75
No; she will still have 75 pages left to read after 10 days.
Module 6
Exercise
Depth of Knowledge (D.O.K.)
Mathematical Processes
3 Strategic Thinking
1.G Explain and justify arguments
2–5
1 Recall of Information
1.C Select tools
6–9
2 Skills/Concepts
1.C Select tools
10–15
2 Skills/Concepts
1.A Everyday life
3 Strategic Thinking
1.A Everyday life
2 Skills/Concepts
1.A Everyday life
1
16
17–18
INTEGRATE MATHEMATICAL
PROCESSES
Focus on Reasoning
Encourage students to check their answers to make
sure they are reasonable. When possible, students
should estimate answers before working out the
problems.
Lesson 2
266
A1_MTXESE346734_U3M06L2 266
© Houghton Mifflin Harcourt Publishing Company
12. Keisha is reading a 325-page book at a rate of 25 pages per day. Use a point-slope equation to determine
whether she will finish reading the book in 10 days.
06/11/14 8:29 PM
Point-Slope Form
266
13. Lizzy is tiling a kitchen floor for the first time. She had a tough time at first and placed only 5 tiles the first
day. She started to go faster, and by the end of day 4, she had placed 35 tiles. She worked at a steady rate
after the first day. Use an equation in point-slope form to determine how many days Lizzy took to place all
of the 100 tiles needed to finish the floor.
AUDITORY CUES
To write the equation of a line in point-slope form,
students need to know the slope and one point on the
line. Students might state the point-slope form as
follows: “For a given point, y minus the y-coordinate
equals the slope times the quantity x minus the
x-coordinate.”
30
35 - 5 _
_
= 10
=
m=
4-1
(100 − 5)
95
95 + 10
105
10.5
3
=10(x − 1)
=10x − 10
=10x − 10 + 10
=10x
=x
Lizzy took 10.5 days to place all the tiles.
14. The amount of fresh water left in the tanks of a
nineteenth-century clipper ship is a linear function of
the time since the ship left port, as shown in the table.
Write an equation in point-slope form that represents the
function. Then find the amount of water that will be left in
the ship’s tanks 50 days after leaving port.
© Houghton Mifflin Harcourt Publishing Company • Image Credits: ©Fine Art
Photographic Library/Corbis
Time (days)
m=
Amount (gal)
1
3555
8
3240
15
2925
3555 - 3240 _
315
__
= -45
=
-7
1-8
y - 3555 = -45(50 - 1)
y - 3555 = -2205
y = 1350
1350 gallons of water will be left after 50 days.
Module 6
Exercise
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Depth of Knowledge (D.O.K.)
Mathematical Processes
2/22/14 4:55 AM
1.A Everyday life
2 Skills/Concepts
1.D Multiple representations
22–23
3 Strategic Thinking
1.G Explain and justify arguments
24
3 Strategic Thinking
1.D Multiple representations
21
Lesson 6.2
Lesson 2
3 Strategic Thinking
19–20
267
267
15. At higher altitudes, water boils at lower temperatures. This relationship between
altitude and boiling point is linear. At an altitude of 1000 feet, water boils at 210 °F. At
an altitude of 3000 feet, water boils at 206 °F. Use an equation in point-slope form to
find the boiling point of water at an altitude of 6000 feet.
AVOID COMMON ERRORS
Remind students to write both the subtraction
symbol and negative symbol when they substitute
negative values in the point-slope form.
206 - 210
-4 = -0.002
m = __ = _
3000 - 1000 2000
y - 210 = -0 . 002(6000 - 1000)
y - 210 = -10
y = 200
The boiling point at 6000 feet is 200 °F.
m=
400 - 175 _
225
_
=
= 45
5
8-3
y - 175 = 45(5 - 3)
500
400
Area (cm2)
16. In art class,Tico is copying a detail from a painting. He paints
slowly for the first few days, but manages to increase his rate
after that. The graph shows his progress after he increased
his rate. How many square centimeters of his painting will he
finish in 5 days after the increase in rate?
y
(8, 400)
300
200
(3, 175)
100
x
y - 175 = 90
0
y = 265
He will finish 265 cm 2 of his painting in 5 days after the
increase in rate.
1 2 3 4 5 6 7 8 9 10
Day since rate increase
© Houghton Mifflin Harcourt Publishing Company
Module 6
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Lesson 2
11/5/14 2:19 PM
Point-Slope Form
268
17. A hot air balloon in flight begins to ascend at a steady rate of 120 feet per
minute. After 1.5 minutes, the balloon is at an altitude of 2150 feet. After
3 minutes, it is at an altitude of 2330 feet. Use an equation in point-slope
form to determine whether the balloon will reach an altitude of 2500 feet in
4 minutes.
INTEGRATE MATHEMATICAL
PROCESSES
Focus on Math Connections
m=
Suggest to students that when they are given two
points, they use both points to write equations in
point-slope form. They can then check that the
equations are equivalent by writing them in
slope-intercept form.
180
2330 - 2150 _
__
= 120
=
3 - 1.5
1.5
y - 2150 = 120(4 - 1.5)
y - 2150 = 300
y = 2450
The balloon will not reach an altitude of 2500 feet in 4 minutes.
18. A candle burned at a steady rate. After 32 minutes, the candle was 11.2 inches tall. Eighteen minutes later,
it was 10.75 inches tall. Use an equation in point-slope form to determine the height of the candle after
2 hours.
-0.45
10.75 - 11.2
=
= -0.025
m= 50 - 32
18
__ _
y - 11.2 = -0.025(120 - 32)
y - 11.2 = -2.2
y=9
© Houghton Mifflin Harcourt Publishing Company • Image Credits: ©RoyaltyFree/Corbis
After 2 hours, the candle will be 9 inches tall.
Module 6
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269
Lesson 6.2
269
Lesson 2
2/22/14 4:55 AM
19. Volume A rectangular swimming pool has a volume of 2160 cubic feet. Water is being added to the pool
at a rate of about 20 cubic feet per minute. Use an equation in point-slope form to determine about how
long it will take to fill the pool completely if there were already about 1200 gallons of water in the pool. Use
the fact that 1 cubic foot of space holds about 7.5 gallons of water.
1200 gal · 1 ft
__
= 160 ft
3
7.5 gal
y - 160
2160 - 160
2000
100
=
=
=
=
COGNITIVE STRATEGIES
Encourage students to look for ways to make
problems easier to solve. For example, in some
problems, any two points can be used to find the
slope, but some pairs may be easier to work with
than others.
3
20(x - 0)
20x
20x
x
It will take about 100 minutes to fill the pool completely.
20. Multi-Step Marisa is walking from her home to her friend Sanjay’s home. When she is 12 blocks away
from Sanjay’s home, she looks at her watch. She looks again when she is 8 blocks away from Sanjay’s home
and finds that 6 minutes have passed.
a. What do you need to assume in order to treat this as a linear situation?
You need to assume that Marisa is walking at a fixed rate.
b. Identify the variables for the linear situation and identify two points on the line.
Explain the meaning of the points in the context of the problem.
x represents the number of minutes since Marisa first looked at her watch, and y
represents the number of blocks she is from Sanjay’s home. The point (0, 12) indicates
that when Marisa first looked at her watch, she was 12 blocks from Sanjay’s home. The
point (6, 8) indicates that 6 minutes after she first looked at her watch she was 8 blocks
from Sanjay’s home.
c. Find the slope of the line and describe what it means in the context of the problem.
m=
8 - 12
2
_
= - _; the slope indicates that for every minute Marisa
6-0
3
2
walks, the distance to Sanjay’s home decreases by _
block.
3
© Houghton Mifflin Harcourt Publishing Company
d. Write an equation in point-slope form for the situation and use it to find the number
of minutes Marisa takes to reach Sanjay’s home. Show your work.
_2 (x - 0)
3
2x
-12 = - _
0 - 12 = 18 = x
3
Marisa takes 18 minutes to reach Sanjay’s home.
Module 6
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Lesson 2
06/11/14 8:31 PM
Point-Slope Form
270
21. Match each equation with the pair of points used to create the equation.
JOURNAL
In their journal, have students compare and contrast
writing linear equations from a point and the slope,
and from two points.
a. y - 10 = 1(x + 2)
c
(0, 0), (-1, 1)
b. y - 0 = 1(x - 0)
b
(1, 1), (-1, -1)
c. y - 3 = -1(x + 3)
a
(-2, 10), (0, 12)
d. y - 3 = 0(x - 2)
d
(1, 3), (-3.5, 3)
H.O.T. Focus on Higher Order Thinking
22. Explain the Error Carlota wrote the equation y + 1 = 2( x – 3 ) for the line passing through the points
( –1, 3 ) and ( 2, 9 ). Explain and correct her error.
Carlota replaced x 1 in the point-slope form with y 1 and vice versa. A correct equation using
the point (-1, 3 ) is y - 3 = 2(x + 1). A correct equation using the point (2, 9) is y - 9 = 2
(x - 2).
23. Communicate Mathematical Ideas Explain why it is possible for a line to have no equation in pointslope form or to have infinitely many, but it is not possible that there is only one.
If the slope is undefined, there is no equation of the line in point-slope form. Otherwise,
there are two equations in point-slope form for every pair of points on the line.
24. Persevere in Problem Solving If you know that A ≠ 0 and B ≠ 0, how can you write an equation in
point-slope form of the equation Ax + By = C ?
To find another point, find the x-intercept,
the value of x when y is 0.
Possible answer: Find the y-intercept by
using the fact that the y-intercept is the
value of y when x is 0.
Ax + By = C
Ax + By = C
© Houghton Mifflin Harcourt Publishing Company
x=
A
C
Then the point (_, 0) is on the line, and
A
an equation in point-slope form
is y - 0 =
A1_MTXESE346734_U3M06L2 271
Lesson 6.2
-By + C
_
A
C
When y = 0, x = _.
_
_
Module 6
271
Ax = -By + C
By = -Ax + C
-Ax + C
y=
B
C
When x = 0, y = .
B
271
-A
_
(x - _C ).
B
A
Lesson 2
06/11/14 8:32 PM
CONNECT CONTEXT
Lesson Performance Task
Some students may be confused by what appear to be
two different uses of the term slope: the mountain has
a constant slope and Alberto snowboards down the
slope. Explain that a slope on a graph is a measure of
how steep the line or hill is. A slope may also mean a
stretch of ground that forms an incline, as on a hill.
Alberto is snow boarding down a mountain with a constant slope. The slope he is on has an
overall length of 1560 feet. The top of the slope has a height of 4600 feet, and the slope has a
vertical drop of 600 feet. It takes him 24 seconds to reach the bottom of the slope.
a. If we assume that Alberto’s speed down the slope is constant,
what is his height above the bottom of the slope at 10 seconds
into the run?
b. Alberto says that he must have been going 50 miles per hour
down the slope. Do you agree? Why or why not?
INTEGRATE MATHEMATICAL
PROCESSES
Focus on Modeling
a. Let h equal height in feet and t equal time in seconds since Alberto left the top of the
slope. The coordinates (t, h) for the points at the top and bottom of the run are (0, 4600)
and (24, 4000).
4600 − 4000
600
Calculate the slope: m = __ = _ = −25
0 − 24
−24
Use the point (0, 4600) and the slope to write a point-slope equation. (Students may
choose the point (24, 4000) and write the equation h − 4000 = −25(t – 24).)
It may help students to make a sketch of the situation
in the Lesson Performance Task. Students should find
that the bottom of the slope is not on the horizontal
axis; it is along h = 4000.
h − h 1 = −25(t − t 1)
h − 4600 = −25(t − 0)
h − 4600 = −25t
h = −25t + 4600
Solve for t = 10:
h = −25t + 4600
h = −25(10) + 4600 = 4350
Module 6
272
© Houghton Mifflin Harcourt Publishing Company
So at 10 seconds, Alberto’s height is at 4350 feet, which is 350 feet above the bottom of
the slope.
1560 ft
b. Alberto’s average speed = _ = 65 ft/sec, which is approximately 44 miles
24 sec
per hour. So students should disagree with Alberto; he was going very fast but not quite
50 miles per hour. Some students might argue that it is unrealistic to assume his speed
was constant: he was probably accelerating as he went down the slope. In that case, his
instantaneous speed could have reached 50 mph even if the average speed was 44 mph.
Lesson 2
EXTENSION ACTIVITY
A1_MTXESE346734_U3M06L2 272
Have students research the length (cm) of a snowboard required for a given rider’s
weight in pounds for several different weights, and then determine if the graph of
(weight, length) appears to be linear.
Students should find that the graph of the (weight, length) points is roughly linear,
although the points do not all lie on the same line.
06/11/14 8:35 PM
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.
Point-Slope Form
272