LESSON
6.2
Name
Transforming
Quadratic Functions
Class
6.2
Date
Transforming Quadratic
Functions
Essential Question: How can you obtain the graph of g(x) = a(x - h) + k from the graph of
f(x) = x 2?
2
Common Core Math Standards
Resource
Locker
The student is expected to:
F-BF.3
Explore
Identify the effect on the graph of replacing f(x) by f(x) + k, kf(x), f(kx),
and f(x + k) for specific values of k (both positive and negative); find the
value of k given the graphs. Experiment with cases and illustrate an
explanation of the effects on the graph using technology. Include
recognizing even and odd functions from their graphs and algebraic
expressions for them. Also F-BF.1, F-BF.4, F-IF.4, F-IF.2
Understanding Quadratic Functions of the Form
2
g(x) = a(x - h) + k
Every quadratic function can be represented by an equation of the form g(x) = a(x - h) + k. The values of the
parameters a, h, and k determine how the graph of the function compares to the graph of the parent function, y = x 2.
2
Use the method shown to graph g(x) = 2(x - 3) + 1 by transforming the graph of ƒ(x) = x 2.
2

Graph ƒ(x) = x 2.
10
Mathematical Practices
8
MP.8 Patterns
6
Language Objective
4
Students work in pairs or small groups to both give and listen to oral clues
about graphs of quadratic functions.


© Houghton Mifflin Harcourt Publishing Company
Possible answer: Identify the vertex (h, k) and use
the sign of a to determine whether the graph opens
up or down. Generate a few points on one side of
the vertex and sketch the graph using those points
and symmetry.
2
Stretch the graph vertically by a factor of 2 to obtain the
graph of y = 2x 2. Graph y = 2x 2.
x
-6
Notice that point (2, 4) moves to point (2, 8) .
ENGAGE
Essential Question: How can you
2
obtain the graph of g(x ) = a(x - h) + k
from the graph of f(x) = x 2?
y
-4
-2
0
-2
2
4
6
-4
Translate the graph of y = 2x 2 right 3 units and up 1 unit
2
to obtain the graph of g(x) = 2(x - 3) + 1. Graph g(x) =
2
(
)
2 x - 3 + 1.
Notice that point (2, 8) moves to point (5, 9) .

(0, 0) while the vertex of the graph of g(x) =
The vertex of the graph of ƒ(x) = x 2 is
2(x - 3) + 1 is
2
(3, 1) .
PREVIEW: LESSON
PERFORMANCE TASK
View the Engage section online. Discuss the photo
and how the path of a ball used in sports can be
modeled by a quadratic function. Then preview the
Lesson Performance Task.
Module 6
be
ges must
EDIT--Chan
DO NOT Key=NL-A;CA-A
Correction
Lesson 2
225
gh “File info”
made throu
Date
Class
Transformi
Functions
6.2
obtain
can you
ion: How 2
f(x) = x ?
Quest
Essential
the full
F-BF.3 For
F-IF.4, F-IF.2
IN2_MNLESE389830_U3M06L2.indd 225
ic
ng Quadrat
Name
Explore
text of these
the graph
standards,
of g(x) =
see the table
)2 + k from
a(x - h
page
starting on
the graph
CA2.Also
of
HARDCOVER PAGES 167176
Resource
Locker
F-BF.1, F-BF.4,
of the
Functions
Form
ic
ing Quadrat
the
values of
Understand- h)2 + k
h) + k. The
y=x .
(x
= a(x function,
form g(x)
g(x) = a
the parent
on of the
graph of
2
2
Turn to these pages to
find this lesson in the
hardcover student
edition.
to the
by an equati
x2 .
n compares
of ƒ(x) =
represented
of the functio
the graph
n can be
tic functio
how the graph 2 1 by transforming
y
Every quadra
k determine ) = 2(x - 3) +
a, h, and
10
g(x
to graph
parameters
d shown
Use the metho
8
2
x) = x .
Graph ƒ(

6
4
2

2 to obtain the
0
of
-2
lly by a factor
-6 -4
-2
graph vertica = 2x2 .
y
Stretch the
2
y = 2x . Graph
(2, 8) .
graph of
-4
to point
4) moves
point (2,
Notice that
unit
and up 1
units
2
3
right 2
g(x) =
of y = 2x
1. Graph
the graph
(x - 3) +
Translate
g(x) = 2
graph of
the
to obtain 2
+ 1.
(5, 9) .
2(x - 3)
to point
of g(x) =
8) moves
point (2,
of the graph
Notice that
(0, 0) while the vertex
x2 is
of ƒ(x) =
of the graph
The vertex
(3, 1) .
2
+ 1 is
2(x - 3)
x
2
4
6

© Houghto
n Mifflin
Harcour t
Publishin
y
g Compan

Lesson 2
225
Module 6
6L2 225
30_U3M0
ESE3898
IN2_MNL
225
Lesson 6.2
09/04/14
6:30 PM
07/04/14 5:35 PM
Reflect
EXPLORE
Discussion Compare the minimum values of ƒ(x) = x 2 and g(x) = 2(x - 3) 2 + 1. How is the minimum
value related to the vertex?
2
The minimum value of f(x) = x 2 is 0 and the minimum value g(x) = 2(x - 3) +1 is 1. The
1.
Understanding Quadratic Functions
2
of the Form g(x ) = a(x - h) + k
minimum value is the y-coordinate of the vertex.
Discussion What is the axis of symmetry of the function g(x) = 2(x - 3) 2 + 1? How is the axis of
symmetry related to the vertex?
2
The axis of symmetry of g(x) = 2(x - 3) + 1 is x = 3. The axis of symmetry always passes
2.
INTEGRATE TECHNOLOGY
through the vertex of the parabola. The x-coordinate of the vertex gives the equation of
Students have the option of completing the activity
either in the book or online.
the axis of symmetry of the parabola.
Explain 1
Understanding Vertical Translations
CONNECT VOCABULARY
A vertical translation of a parabola is a shift of the parabola up or down, with no change in the shape of the parabola.
This lesson discusses translation in terms of a
transformation of a function graph. English learners
may know about language translation. Discuss with
students how the two meanings of translate are
different.
Vertical Translations of a Parabola
The graph of the function ƒ(x) = x + k is the graph of ƒ(x) = x translated vertically.
2
2
If k > 0, the graph ƒ(x) = x 2 is translated k units up.
If k < 0, the graph ƒ(x) = x 2 is translated ⎜k⎟ units down.
Example 1

Graph each quadratic function. Give the minimum or maximum value
and the axis of symmetry.
g(x) = x 2 + 2
10
Make a table of values for the parent function f(x) = x 2
and for g(x) = x 2 + 2. Graph the functions together.
8
f(x) = x 2
g(x) = x 2 + 2
6
-3
9
11
4
-2
4
6
-1
1
3
0
0
2
1
1
3
4
6
3
9
11
f(x) = x2
2
x
-6
-4
-2
0
-2
2
4
6
-4
Understanding Vertical Translations
© Houghton Mifflin Harcourt Publishing Company
x
2
EXPLAIN 1
y
QUESTIONING STRATEGIES
How is the graph of g(x) = x 2 + 2 related to
the graph of g(x) = x 2 – 5? Both are
translated graphs of the same parent function,
f(x) = x 2, but g(x) = x 2 + 2 is translated 2 units up
and g(x) = x 2 - 5 is translated 5 units down. So, the
graph of g(x) = x 2 + 2 is 7 units higher than the
graph of g(x) = x 2 - 5.
The function g(x) = x 2 + 2 has a minimum value of 2.
The axis of symmetry of g(x) = x 2 + 2 is x = 0.
Module 6
Is the vertex of the graph of g(x) = x 2 + 2
the same as the vertex of the graph of
g(x) = x 2 – 5? No; g(x) = x 2 + 2 has vertex (0, 2),
and g(x) = x 2 - 5 has vertex (0, -5).
Lesson 2
226
PROFESSIONAL DEVELOPMENT
IN2_MNLESE389830_U3M06L2 226
18/07/14 7:48 PM
Math Background
In this lesson, students graph the family of quadratic functions of the form
2
g(x) = a(x - h) + k and compare those graphs to the graph of the parent
function f (x) = x 2. Some key understandings are:
• The function f (x) = x 2 is the parent of the family of all quadratic functions.
• To graph a quadratic function of the form g(x) = a(x - h) + k, identify the
vertex (h, k). Then determine whether the graph opens upward or downward.
Then generate points on either side of the vertex and sketch the graph of the
function.
2
Transforming Quadratic Functions
226
B
g(x) = x - 5
2
Make a table of values for the parent function ƒ(x) = x
and for g(x) = x 2 - 5. Graph the functions together.
y
10
2
x
f(x) = x 2
g(x) = x 2 - 5
-3
9
4
-2
4
8
6
4
2
-1
-1
1
-4
0
0
-5
1
1
-4
2
4
-1
3
9
4
x
-6
-4
-2
0
2
-2
4
6
-4
-6
The function g(x) = x 2 - 5 has a minimum value of -5.
The axis of symmetry of g(x) = x 2 - 5 is x = 0 .
© Houghton Mifflin Harcourt Publishing Company
Reflect
3.
How do the values in the table for g(x) = x 2 + 2 compare with the values in the table for the parent
function ƒ(x) = x 2?
For each x in the table, g(x) is 2 greater than f(x).
4.
How do the values in the table for g(x) = x 2 - 5 compare with the values in the table for the parent
function ƒ(x) = x 2?
For each x in the table, g(x) is 5 less than f(x).
Your Turn
Graph each quadratic function. Give the minimum or maximum value and the axis of symmetry.
5.
g(x) = x 2 + 4
y
8
The function g(x) = x 2 + 4 has a minimum value of 4.
The axis of symmetry for g(x) = x + 4 is x = 0.
2
6
4
2
x
-4
Module 6
227
-2
0
2
4
Lesson 2
COLLABORATIVE LEARNING
IN2_MNLESE389830_U3M06L2.indd 227
Peer-to-Peer Activity
Have students work in pairs. Have one student draw a graph of y = x 2 + k for
some value of k. The second student then writes the equation for the graph.
Students then compare their results and determine whether the equation is correct
for the given graph. Have students take turns in the two roles.
227
Lesson 6.2
4/2/14 12:37 AM
g(x) = x 2 - 7
6.
y
The function g(x) = x - 7 has a minimum value of -7.
2
EXPLAIN 2
2
The axis of symmetry for g(x) = x 2 - 7 is x = 0.
x
-4
-2
0
2
-2
4
Understanding Horizontal
Translations
-4
-6
QUESTIONING STRATEGIES
How is the graph of g(x) = (x – 1) related to
2
the graph of g(x) = (x + 2) ? Both are
translated graphs of the same parent function,
2
f(x) = x 2 , but the graph of g(x) = (x - 1) is
translated 1 unit to the right and has vertex (1, 0),
2
while the graph of g(x) = (x + 2) is translated
2 units to the left and has vertex (-2, 0). So, the
2
graph of g(x) = (x - 1) is 3 units to the right of the
2
graph of g(x) = (x + 2) .
2
Understanding Horizontal Translations
Explain 2
A horizontal translation of a parabola is a shift of the parabola left or right, with no change in the shape of the
parabola.
Horizontal Translations of a Parabola
The graph of the function ƒ(x) = (x – h) is the graph of ƒ(x) = x 2 translated horizontally.
2
If h > 0, the graph ƒ(x) = x 2 is translated h units right.
If h < 0, the graph ƒ(x) = x 2 is translated h units left.
Example 2

Graph each quadratic function. Give the minimum or maximum value and the axis of
symmetry.
g(x) = (x - 1)
2
Make a table of values for the parent function ƒ(x) = x
2
and for g(x) = (x - 1) . Graph the functions together.
10
What is the vertex of the graph of
2
g(x) = (x – h) ? (h, 0)
y
2
8
f(x) = x 2
g(x) = (x -1)
6
-3
9
16
4
-2
4
9
2
-1
1
4
0
0
1
1
1
0
-2
2
4
1
-4
3
9
2
x
-6
-4
0
-2
2
4
6
4
The function g(x) = (x - 1) has a minimum value of 0.
2
© Houghton Mifflin Harcourt Publishing Company
x
The axis of symmetry of g(x) = (x - 1) is x = 1.
2
Module 6
Lesson 2
228
DIFFERENTIATE INSTRUCTION
IN2_MNLESE389830_U3M06L2 228
18/07/14 7:48 PM
Visual Cues
Have students take a coordinate grid and label it “Vertex of g(x) = (x – h) + k.”
Have them place these points, labels, and functions into the four quadrants.
2
(2, 3)
(-2, 3)
(–2, –3)
(2, –3)
h = 2, k = 3
g(x) = (x - 2) + 3
2
h = -2, k = 3
g(x) = (x + 2) + 3
h = 2, k = –3
g(x) = (x – 2) – 3
h = –2, k = –3
2
g(x) = (x + 2) – 3
2
2
Students can use this graph as a reminder of how the location of the vertex and
the function are related.
Transforming Quadratic Functions
228
B
AVOID COMMON ERRORS
g(x) = (x +1)
2
10
Make a table of values and graph the functions together.
Students may forget that they can use a pattern to
write equations from graphs. Remind students that
adding k to x 2 moves the graph up for k > 0 or down
for k < 0 and that subtracting h from x moves the
graph left for h < 0 or right for h > 0. This is true for
all nonzero values of k and h.
x
f(x) = x 2
g(x) = (x + 1)
-3
9
4
-2
4
1
-1
1
0
0
0
1
1
1
4
2
4
9
3
9
16
y
8
2
6
4
2
x
-6
-4
-2
0
2
-2
4
6
-4
2
The function g(x) = (x +1) has a minimum value of 0 .
2
The axis of symmetry of g(x) = (x + 1) is x = -1 .
© Houghton Mifflin Harcourt Publishing Company
Reflect
7.
How do the values in the table for g(x) = (x - 1) compare with the values in the table for the parent
function ƒ(x) = x 2?
For each x in the table, g(x) is the same as f(x - 1).
8.
How do the values in the table for g(x) = (x + 1) compare with the values in the table for the parent
function ƒ(x) = x 2?
For each x in the table, g(x) is the same as f(x + 1).
2
2
Your Turn
Graph each quadratic function. Give the minimum or maximum value and the
axis of symmetry.
9.
g(x) = (x - 2)
10. g(x) = (x + 3)
2
2
y
y
16
16
12
12
8
8
4
4
x
x
-6
-4
-2
0
2
4
Module 6
-6
6
229
-4
-2
0
2
4
6
Lesson 2
LANGUAGE SUPPORT
IN2_MNLESE389830_U3M06L2.indd 229
Communicate Math
Have each student sketch a graph of a parabola on a card, and write a quadratic
function in any form on another card. Then have them write clues about the graph
and about the function. For example, “The parabola opens upward/downward. Its
axis of symmetry is _____; the vertex is at the point _____. The function’s graph
will open downward/upward.” Provide sentence stems if needed to help students
begin their clues. Collect the graph and function cards in one pile, and the clue
cards in another. Have other students match graph and function cards to fit the
clues.
229
Lesson 6.2
4/2/14 12:37 AM
Graphing g(x) = a(x - h) + k
2
Explain 3
EXPLAIN 3
The vertex form of a quadratic function is g(x) = a(x - h) + k, where the point (h, k) is the vertex. The axis of
symmetry of a quadratic function in this form is the vertical line x = h.
2
Graphing g(x) = a(x - h) + k
To graph a quadratic function in the form g(x) = a(x - h) + k, first identify the vertex ( h, k ). Next, consider the
sign of a to determine whether the graph opens upward or downward. If a is positive, the graph opens upward. If a is
negative, the graph opens downward. Then generate two points on each side of the vertex. Using those points, sketch
the graph of the function.
2
2
Example 3

QUESTIONING STRATEGIES
Graph each quadratic function.
g(x) = -3(x + 1) - 2
y
2
-6
Identify the vertex.
-4
The vertex is at (-1, -2).
Make a table for the function. Find two points on each side
of the vertex.
-2
0
2
-2
What can you tell about the graph of a
function from its equation in the form
2
g(x) = a(x - h) + k? the location of the vertex
and whether the graph opens upward or downward
x
4
6
-4
-6
What are the domain and range for a
quadratic function whose graph opens
downward? The domain is all real numbers, and the
range is the set of all values less than or equal to the
maximum value.
-8
x
-3
-2
-1
0
1
g(x)
-14
-5
-2
-5
-14
Plot the points and draw a parabola through them.
-10
-12
-14
-16

AVOID COMMON ERRORS
g(x) = 2(x - 1) - 7
2
Students may try to graph a quadratic function of the
2
form g(x) = a(x - h) + k by using a value other
than x = h. Remind them that they need to first
identify and plot the vertex. Then they should
identify and plot other points and use the plotted
points to draw a parabola.
y
15
Identify the vertex.
The vertex is at (1, -7) .
12
9
6
x
-2
0
1
2
4
g(x)
11
-5
-7
-5
11
3
Plot the points and draw a parabola through them.
x
-3
0
-3
3
-6
© Houghton Mifflin Harcourt Publishing Company
Make a table for the function. Find two points on each side of the vertex.
INTEGRATE MATHEMATICAL
PRACTICES
Focus on Modeling
MP.4 Tell students that a transformed quadratic
Reflect
11. How do you tell from the equation whether the vertex is a maximum value or a minimum value?
If the value of a is positive, the vertex is a minimum value. If the value of a is negative, the
vertex is a maximum value.
Module 6
IN2_MNLESE389830_U3M06L2.indd 230
230
Lesson 2
21/07/14 3:31 PM
function models the height of an object dropped
from a given height, based upon the time since it was
dropped. Sketch a quadratic function that models the
situation, and draw students’ attention to the vertex
(0, k) being the maximum point of the graph. Ask
about the sign of a in the function g(x) = ax 2 + k, and
note that the values to the left of the y-axis are not
considered.
Transforming Quadratic Functions
230
Your Turn
ELABORATE
Graph each quadratic function.
13. g(x) = 2(x + 3) - 1
12. g(x) = -(x - 2) + 4
2
2
INTEGRATE MATHEMATICAL
PRACTICES
Focus on Technology
MP.5 Give students a function in the form
x
-1
0
2
4
5
x
-5
-4
-3
-2
-1
g(x)
-5
0
4
0
-5
g(x)
7
1
-1
1
7
g(x) = a(x – h) + k. Have students use the
whiteboard to identify and plot the vertex and then
identify and plot other points on the graph before
drawing the graph of the function.
2
6
y
8
6
4
4
2
x
-6
-4
-2
SUMMARIZE
0
-2
2
4
2
6
x
-6
-4
How do you graph a quadratic function of the
2
form g(x ) = a(x – h) + k? First, identify and
plot the vertex. Then, identify and plot other points
on the graph. Finally, draw the graph.
y
-4
-2
0
-2
2
4
6
-4
-6
Elaborate
14. How does the value of k in g(x) = x 2 + k affect the translation of ƒ(x) = x 2?
If k > 0, the graph f(x) = x 2 is translated k units up.
If k < 0, the graph f(x) = x 2 is translated k units down.
15. How does the value of h in g(x) = (x - h) affect the translation of ƒ(x) = x 2?
If h > 0, the graph f(x) = x 2 is translated h units right.
© Houghton Mifflin Harcourt Publishing Company
2
If h < 0, the graph f(x) = x 2 is translated h units left.
16. In g(x) = a(x - h) + k, what are the coordinates of the vertex?
(h, k)
2
17. Essential Question Check-In How can you use the values of a, h, and k, to obtain the graph of
2
g(x) = a(x - h) + k from the graph ƒ(x) = x 2?
The graph of f(x) = x 2 is stretched or compressed by a factor of ⎜a⎟, and reflected across the
x-axis if a is negative; it is translated h units horizontally and k units vertically.
Module 6
IN2_MNLESE389830_U3M06L2.indd 231
231
Lesson 6.2
231
Lesson 2
21/07/14 3:38 PM
Evaluate: Homework and Practice
EVALUATE
• Online Homework
• Hints and Help
• Extra Practice
Graph each quadratic function by transforming the graph of ƒ(x) = x 2. Describe
the transformations.
1.
g(x) = 2(x - 2) + 5
2
16
2.
g(x) = 2(x + 3) - 6
2
y
14
6
12
4
10
2
-8
6
-6
-4
0
-2
-2
x
0
2
4
6
The parent function has been translated
3 units left and 6 units down. It has been
stretched vertically by a factor of 2.
1 (x - 3) - 4
g(x) = _
2
2
4.
8
y
10
-2
6
8
4
6
0
-2
4
2
4
6
2
8
x
-6
-4
-4
-2
-6
Exercise
IN2_MNLESE389830_U3M06L2 232
Exercises 1–4
Example 1
Understanding Vertical Translations
Exercises 5–10
Example 2
Understanding Horizontal
Translations
Exercises 11–16
Example 3
2
Graphing g(x) = a(x - h) + k
Exercises 17–27
-2
2
4
6
8
k, and a come from. Give coordinates for a vertex and
have students substitute the x- and y-values of the
vertex into the equation of h and k, determine the
value of a, and then write the equation of the
function.
The parent function has been translated
4 units right and 2 units down. It has been
stretched vertically by a factor of 3.
The parent function has been translated
3 units right and 4 units down. It has been
1
.
vertically compressed by a factor of __
2
Module 6
0
-4
-8
Explore
Understanding Quadratic Functions
2
of the Form g(x ) = a(x - h) + k
INTEGRATE MATHEMATICAL
PRACTICES
Focus on Modeling
MP.4 Make sure that students understand where h,
y
© Houghton Mifflin Harcourt Publishing Company
-4
Practice
2
x
-6
Concepts and Skills
g(x) = 3(x - 4) - 2
2
-8
6
-8
The parent function has been translated
2 units right and 5 units up. It has been
stretched vertically by a factor of 2.
3.
4
-6
2
-2
2
-4
4
-4
ASSIGNMENT GUIDE
x
8
-6
y
8
Lesson 2
232
Depth of Knowledge (D.O.K.)
Mathematical Practices
1–8
1 Recall of Information
MP.6 Precision
9–12
2 Skills/Concepts
MP.5 Using Tools
13–16
2 Skills/Concepts
MP.2 Reasoning
17–20
2 Skills/Concepts
MP.6 Precision
21–22
2 Skills/Concepts
MP.2 Reasoning
23–25
3 Strategic Thinking
MP.3 Logic
26–27
3 Strategic Thinking
MP.2 Reasoning
18/07/14 7:48 PM
Transforming Quadratic Functions
232
Graph each quadratic function.
VISUAL CUES
5.
g(x) = x 2 - 2
Have students create a design made of transformed
parabolas and keep a record of each parabola’s
function. Encourage students to use their
imaginations to add colors and patterns to the design.
6.
g(x) = x 2 + 5
y
y
6
8
4
6
2
4
x
-4
INTEGRATE MATHEMATICAL
PRACTICES
Focus on Technology
MP.5 Allow students to use graphing
7.
calculators to check their work. Some students
will be motivated to explore additional types of
transformations.
-2
0
2
-2
2
4
x
-4
g(x) = x 2 - 6
8.
2
-2
-4
-2
-2
y
4
2
4
y
8
2
4
6
4
-4
2
-6
x
-4
-8
9.
2
g(x) = x 2 + 3
x
0
0
-2
0
Graph g(x) = x 2 - 9. Give the minimum or maximum value and the axis of symmetry.
y
© Houghton Mifflin Harcourt Publishing Company
The function has a minimum value of -9.
The axis of symmetry is x = 0.
-4
-2
0
-2
x
2
4
-4
-6
-8
10. How is the graph of g(x) = x 2 + 12 related to the graph of ƒ(x) = x 2?
The graph of g(x) = x 2 + 12 is the graph of f(x) = x 2 translated 12 units up.
Module 6
IN2_MNLESE389830_U3M06L2.indd 233
233
Lesson 6.2
233
Lesson 2
4/2/14 12:36 AM
Graph each quadratic function. Give the minimum or maximum
value and the axis of symmetry.
AVOID COMMON ERRORS
11. g(x) = (x - 3)
Some students may automatically say that the
function has a minimum when a parabola opens
downward, and a maximum when a parabola opens
upward, because of word association. Tell students to
visualize the graph before determining whether it has
a minimum or maximum.
2
y
36
30
24
18
12
6
x
-8
-6
-4
0
-2
2
4
6
8
10
The function has a minimum value of 0.
The axis of symmetry is x = 3.
12. g(x) = (x + 2)
2
28
y
24
20
16
© Houghton Mifflin Harcourt Publishing Company
12
8
4
x
-8
-6
-4
0
-2
2
4
6
8
The function has a minimum value of 0.
The axis of symmetry is x = -2.
13. How is the graph of g(x) = (x + 12) related to the graph of ƒ(x) = x 2?
2
The graph of g(x) = (x + 12) is the graph of f(x) = x 2 translated 12 units left.
2
14. How is the graph of g(x) = (x - 10) related to the graph of ƒ(x) = x 2?
2
The graph of g(x) = (x - 10) is the graph of f(x) = x 2 translated 10 units right.
2
Module 6
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Lesson 2
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Transforming Quadratic Functions
234
15. Compare the given graph to the graph of the parent
function ƒ(x) = x 2. Describe how the parent function
must be translated to get the graph shown here.
KINESTHETIC EXPERIENCE
Display each function below, one at a time. Have
students discuss, in pairs, whether to lift their arms
up in the shape of a U to signal the graph opens
upward, or move them downward in the shape of an
upside-down U, to signal that the graph opens
downward. Then have students demonstrate their
decisions.
8
y
6
Translate the graph of the parent function 2 units
to the right.
4
2
x
-4
-2
0
2
-2
4
-4
y = -3x 2 + 18 down
16. For the function g(x) = (x - 9) give the minimum or maximum value and the axis of symmetry.
2
y = 5x + 8 - __15 x 2 down
The minimum value is 0.
The axis of symmetry is x = 9.
-2x 2 +y = -5 up
Graph each quadratic function. Give the minimum or maximum value and the axis of
symmetry.
3x - y = -x 2 up
17. g(x) = (x - 1) - 5
18. g(x) = -(x + 2) + 5
2
2
y
6
y
6
4
4
2
2
x
© Houghton Mifflin Harcourt Publishing Company
-6
-2
2
-2
4
x
-6
6
-4
-2
0
2
-2
-4
-4
-6
-6
4
6
-2
0
1
3
4
x
-5
-3
-2
0
1
g(x)
4
-4
-5
-1
4
g(x)
-4
4
5
1
-4
The function has a minimum value of -5.
The function has a maximum value of 5.
The axis of symmetry is x = 1.
IN2_MNLESE389830_U3M06L2 235
Lesson 6.2
-4
0
x
Module 6
235
6
The axis of symmetry is x = -2.
235
Lesson 2
18/07/14 7:48 PM
1 (x + 1) - 7
19. g(x) = _
4
1 (x + 3) 2 + 8
20. g(x) = -_
3
2
6
y
10
4
6
x
-6
-4
-2
0
-2
Have students work in pairs. Have each student
change one or more of the parameters in
2
f(x) = a(x – h) + k and graph the function. Then
have students trade graphs and try to write the
function equation for the other student’s graph. Have
each student justify the function equation and discuss
it with the partner.
8
2
-8
PEERTOPEER ACTIVITY
y
2
4
4
6
2
x
-4
-12 -10 -8
-6
-6
-4
-2
-8
0
2
-2
4
6
-4
-6
x
-8
-5
-1
3
5
x
-7
-6
-3
0
3
g(x)
5.25
-3
-7
-3
2
g(x)
2.67
5
8
5
-4
The function has a minimum value of -7.
The function has a maximum value of 8.
The axis of symmetry is x = -1.
The axis of symmetry is x = -3.
21. Compare the given graph to the graph of the parent function ƒ(x) = x 2.
Describe how the parent function must be translated to get the graph
shown here.
8
y
6
Translate the graph of the parent function 3 units to the
right and 2 units up.
4
2
0
-2
2
4
6
22. Multiple Representations If the graph of ƒ(x) = x 2 is translated 11 units to the
left and 5 units down, which function is represented by the translated graph?
a. g(x) = (x - 11) - 5
2
b. g(x) = (x + 11) - 5
2
c. g(x) = (x + 11) + 5
2
d. g(x) = (x - 11) + 5
2
© Houghton Mifflin Harcourt Publishing Company
x
-2
e. g(x) = (x - 5) - 11
2
f. g(x) = (x - 5) + 11
2
g. g(x) = (x + 5) - 11
2
h. g(x) = (x + 5) + 11
2
Module 6
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Lesson 2
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Transforming Quadratic Functions
236
JOURNAL
H.O.T. Focus on Higher Order Thinking
Critical Thinking Use a graphing calculator to compare the graphs of y = (2x) 2,
2
2
y = (3x) , and y = (4x) with the graph of the parent function y = x2. Then compare
In their journals, have students explain how to use
the values of a, h, and k to obtain the graph of
2
g(x) = a(x – h) + k from the graph of f(x) = x 2.
( )
( )
( )
2
2
2
the graphs of y = __12 x , y = __13 x , and y = __14 x with the graph of the parent
2
function y = x .
23. Explain how the parameter b horizontally stretches or compresses the graph of
2
y = (bx) when b > 1.
When b > 1, the graph of y = (bx) is compressed horizontally by a factor
1
.
of __
2
b
24. Explain how the parameter b horizontally stretches or compresses the graph of
2
y = (bx) when 0 < b < 1.
When 0 < b < 1, the graph of y = (bx) is stretched horizontally by a
1
.
factor of __
2
b
25. Explain the Error Nina is trying to write an equation for the function represented
by the graph of a parabola that is a translation of ƒ(x) = x 2. The graph has
been translated 4 units to the right and 2 units up. She writes the function as
2
g(x) = (x + 4) + 2. Explain the error.
Nina should have subtracted 4 from x in the equation instead of adding it.
26. Multiple Representations A group of engineers drop an experimental tennis ball
from a catwalk and let it fall to the ground. The tennis ball’s height above the ground
2
(in feet) is given by a function of the form ƒ(t) = a(t - h) + k where t is the time (in
seconds) after the tennis ball was dropped. Use the graph to find the equation for ƒ(t).
32
y
(0, 30)
© Houghton Mifflin Harcourt Publishing Company
24
16
(1, 14)
8
t
0
1
2
3
4
2
f(t) = a(t - 0) + 30, or f(t) = at 2 + 30
14 = a(1) + 30
2
-16 = a
The equation for the function is f(t) = -16t 2 + 30.
27. Make a Prediction For what values of a and c will the graph of ƒ(x) = ax 2 + c have
one x-intercept?
For any real value of a with a ≠ 0, the function will have one x-intercept when c = 0.
Module 6
IN2_MNLESE389830_U3M06L2 237
237
Lesson 6.2
237
Lesson 2
7/22/14 11:39 AM
Lesson Performance Task
Baseball’s Height
The path a baseball takes after it has been hit is modeled by the graph.
The baseball’s height above the ground is given by a function of the form
2
ƒ(t) = a(t - h) + k, where t is the time in seconds since the baseball
was hit.
b. When does the baseball hit the ground?
y
192
Height (ft)
a. What is the baseball’s maximum height? At what time was
the baseball at its maximum height?
INTEGRATE MATHEMATICAL
PRACTICES
Focus on Patterns
MP.8 Discuss with students which type of hit—a
ground ball, a pop-up, or a line drive—would likely
make a baseball have the path shown in the graph.
Ask how knowing the maximum height of the ball
and the time it takes the ball to hit the ground help
you write an equation to represent the path of the
baseball.
The highest height of the ball, the vertex of the
path, is the ordered pair (h, k), and the time it takes
the ball to hit the ground is the ordered pair (t, f(t)),
so the h, k, t, and f(t) values can be substituted into
2
the standard form f(t) = a(t - h) + k to find the
value of a.
144
96
48
c. Find an equation for ƒ(t).
x
d. A player hits a second baseball. The second baseball’s path is
2
modeled by the function g(t) = -16(t - 4) + 256. Which
baseball has a greater maximum height? Which baseball is
in the air for the longest?
0
2
4
6
Time (s)
8
a. The vertex of the parabola is (3, 144). So, the baseball is at its
maximum height of 144 feet after 3 seconds.
b. The second x-intercept of the graph is (6, 0). So, the baseball hits the
ground after 6 seconds.
c. The vertex of the parabola is (3, 144) and one intercept of the graph is
(6, 0). Solve for a.
f(t) = a(t - h) + k
2
INTEGRATE MATHEMATICAL
PRACTICES
Focus on Critical Thinking
MP.3 Point out that the model students write in the
0 = a(6 - 3) + 144
2
-144 = 9a
-16 = a
2
g(t) = -16(t-4) + 256
0 = -16(t-4) + 256
2
-256 = -16(t-4)
16 = (t-4)
√―
16 = t-4
form f(t) = a(t - h) + k to represent the path the
baseball takes can be used to approximate the height
h in feet above the ground after t seconds because it
does not account for air resistance, wind, or other
real-world factors.
2
© Houghton Mifflin Harcourt Publishing Company
2
So, f(t) = -16(t - 3) + 144.
d. The vertex is (4,256) so the baseball was at its maximum height of
256 feet after 4 seconds.
2
2
±4 + 4 = t
0, 8 = t
The ball hits the ground after 8 seconds.
So, the second baseball has a greater maximum height and it traveled
longer in the air than the first.
Module 6
238
Lesson 2
EXTENSION ACTIVITY
IN2_MNLESE389830_U3M06L2.indd 238
Have groups of students draw or tape a large, first-quadrant coordinate grid on
the chalkboard. Have one student toss a tennis ball in front of the grid, making
sure that the path of the ball stays within the grid’s borders, while another student
videotapes the toss at a rate of about 15 frames per second. Then have students
play back the video, marking points on the grid to show the path of the ball.
2
Finally, have students use the model f(t) = a(t - h) + k to write a function
that models the path of the tennis ball. Students may discover that the angle at
which the ball is tossed affects the height and width of the curved path the ball
follows. Have students save their data for Part 2 of the Extension Activity in the
following lesson.
4/2/14 12:36 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.
Transforming Quadratic Functions
238