Transformations of function graphs explore how various parameters affect the shape and position of a function's graph. This resource delves into horizontal and vertical translations, stretches, compressions, and reflections, providing a comprehensive understanding for high school mathematics students. It includes examples, input-output tables, and detailed explanations of how changes in function equations impact their graphical representations. Ideal for students preparing for advanced math exams or those seeking to enhance their understanding of function transformations.

Key Points

  • Explains horizontal and vertical translations of function graphs.
  • Covers vertical stretches and compressions with examples.
  • Includes reflections across the x-axis and y-axis.
  • Provides input-output tables for practical understanding.
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© Houghton Mifflin Harcourt Publishing Company
Explore 1
Investigating Translations
of Function Graphs
You can transform the graph of a function in various ways. You can translate
the graph horizontally or vertically, you can stretch or compress the graph
horizontally or vertically, and you can reflect the graph across the x-axis or the
y-axis. How the graph of a given function is transformed is determined by the
way certain numbers, called parameters, are introduced in the function.
The graph of ƒ
(
x
)
is shown. Copy this graph and use the same grid for the
exploration.
First graph g
(
x
)
= ƒ
(
x
)
+ k where k is the parameter. Let k = 4 so
that g
(
x
)
= ƒ
(
x
)
+ 4. Complete the input-output table and then graph g(x). In general, how
is the graph of g
(
x
)
= ƒ
(
x
)
+ k related to the graph of ƒ
(
x
)
when k is a positive number?
xf
(
x
)
f
(
x
)
+ 4
-1 -22
126
3 -2
52
Now try a negative value of k in g
(
x
)
= ƒ
(
x
)
+ k.
Let k = -3 so that g
(
x
)
= ƒ
(
x
)
- 3. Complete the input-
output table and then graph g
(
x
)
on the same grid.
In general, how is the graph of g
(
x
)
= ƒ
(
x
)
+ k related
to the graph of ƒ
(
x
)
when k is a negative number?
xf(x) f(x) - 3
-1 -2 -5
12-1
3 -2
52
Resource
Locker
?
?
?
?
-
4
-
6
4
6
2
0
y
462
x
-
2
-
4
-
6
f(x) + 4
f(x) - 3
-
4
-
6
4
6
2
y
0
462
x
-
2
-
4
-
6
2
See graph below.
A–B.
6
-5
-1
For k < 0, the graph of g(x) = f(x) + k is
the graph of f(x) translated down
⎜
k
⎟
units.
For k > 0, the graph of g(x) = f(x) + k is
the graph of f(x) translated up k units.
Module 1
23
Lesson 3
1.3 Transformations
of FunctionGraphs
Essential Question: What are the ways you can transform the graph of the function f(x)?
Transformations of
Function Graphs
Online Resources
An extra example for each Explain section is
available online.
Engage
Essential Question: What are the ways you can
transform the graph of the function y = f
(
x
)
?
Possible answer: The parameter h in y =f
(
x - h
)
produces a horizontal translation of the graph
of y = f
(
x
)
. The parameter k in y = f
(
x
)
+ k produces a
vertical translation of the graph of y = f
(
x
)
. The
parameter a in y = af
(
x
)
produces a vertical stretch/
compression of the graph of y = f
(
x
)
and may also
produce a reflection across the x-axis. The parameter
b in y = f
(
1
_
b
x
)
produces a horizontal stretch/
compression of the graph of y = f
(
x
)
and may also
produce a reflection across the y-axis.
Preview: Lesson Performance Task
View the online Engage. Discuss the photo and
the guidelines someone might need to follow
when designing a chair. Then preview the Lesson
Performance Task.
LESSON
1.3
Professional Development
Math Background
Transformations change the graph of a function. When students understand the
basic transformations (translation, reflection, stretch, and compression), they are
better able to understand how to write the equation of a graph, and how to
identify the graph of a function that has been transformed.
Learning Objective
Students will investigate translations, stretches,
compressions, and reflections of function graphs, and
transform the graph of the parent quadratic function.
Math Processes and Practices
MPP2 Abstract and Quantitative Reasoning
Language Objective
Identify graphs of odd and even functions and justify
reasoning with a partner.
Lesson 1.3 23
© Houghton Mifflin Harcourt Publishing Company

Now graph g
(
x
)
= ƒ
(
x - h
)
where h is the parameter. Let h = 2 so that g
(
x
)
= ƒ
(
x - 2
)
.
Complete the mapping diagram and then graph g
(
x
)
with f(x). (Tocomplete the mapping
diagram, you need to find the inputs for g that produce the inputs for ƒ after you subtract 2.
Work backward from the inputs for ƒ to the missing inputs for gby adding 2.) In general,
how
is the graph of g
(
x
)
= ƒ
(
x - h
)
related to the graph of ƒ
(
x
)
whenhis a positive number?
Make a Conjecture How would you expect the graph of g
(
x
)
= ƒ
(
x - h
)
to be related to
the graph of ƒ
(
x
)
when h is a negative number?
Reflect
1. Suppose a function ƒ
(
x
)
has a domain of
⎡
⎣
x
1
, x
2
⎤
⎦
and a range of
⎡
⎣
y
1
, y
2
⎤
⎦
. When the graph of ƒ
(
x
)
is
translated vertically k units where k is either positive or negative, how do the domain and range change?
2. Suppose a function ƒ
(
x
)
has a domain of
⎡
⎣
x
1
, x
2
⎤
⎦
and a range of
⎡
⎣
y
1
, y
2
⎤
⎦
. When the graph of ƒ
(
x
)
is
translated horizontally h units where h is either positive or negative, how do the domain and range change?
3. You can transform the graph of ƒ
(
x
)
to obtain the graph of g
(
x
)
= ƒ
(
x - h
)
+ k by combining
transformations. Predict what wi ll happen by completing the table.
Sign of h Sign of k Transformations of the Graph of f(x)
++Translate right h units and up k units.
+-
-+
--
?
?
?
5
7
Input
for g
Input
for f
Output
for f
Output
for g
1
3
-
1
1
3
5
-
2
-
2
2
-
2
2
-
2
2
-
2
2
?
?
-
4
-
6
4
6
2
0
462
x
-
2
-
4
-
6
For h > 0, the graph of g(x) = f(x - h) is the graph of f(x)
translated right h units.
For h < 0, the graph of g(x) = f(x - h) is the graph of f(x) translated left
⎜
h
⎟
units.
See below.
1. The domain remains the same, and the range changes from
⎡
⎣
y
1
, y
2
⎤
⎦
to
⎡
⎣
y
1
+ k, y
2
+ k
⎤
⎦
.
2. The domain changes from
⎡
⎣
x
1
, x
2
⎤
⎦
to
⎡
⎣
x
1
- h, x
2
- h
⎤
⎦
, and the range remains the same
.
h +, k -: Translate right h units and down
⎜
k
⎟
units.
h -, k +: Translate left
⎜
h
⎟
units and up k units.
h -, k -: Translate left
⎜
h
⎟
units and down
⎜
k
⎟
units.
Module 1
24
Lesson 3
Explore 1
Investigating Translations of
Function Graphs
Integrate Technology
Students have the option of completing the activity
either in the book or online.
Avoid Common Errors
Students may confuse directions in horizontal
translations. Emphasize that h is the number
subtracted from x. For example, in f
(
x – 3
)
, the value of
h is 3, a positive number, and therefore the translation
is to the right.
Questioning Strategies
Given the graph of a function f
(
x
)
, and the graph of
the image of the function after a translation, how
can you determine the rule for the function
represented by the image? You can select a
particular point on the original graph (such as an
endpoint of a segment, or a local maximum or
minimum point), and see how its image was
obtained. If the image is above or below the
original point, the rule will involve adding or
subtracting the number of units it was translated
to f
(
x
)
. If the image is to the left or right of the
original point, the rule will involve adding or
subtracting the number of units it was translated
to x in f
(
x
)
.
Collaborative Learning
Peer-to-Peer Activity
Have students work in pairs. Provide students with three or four functions, and
have them use a graphing calculator to explore how changes to the various
parameters in each function affect the graph of the function. Once they start to
make the connections, encourage them to try and predict each change before
graphing the transformation.
24 Transformations of Function Graphs
© Houghton Mifflin Harcourt Publishing Company
Explore 2
Investigating Stretches and Compressions
of Function Graphs
In this activity, you will consider what happens when you multiply by a positive parameter inside or outside a
function. Throughout, you will use the same function ƒ
(
x
)
that you used in the previous activity.
First graph g
(
x
)
= a ⋅ ƒ
(
x
)
where a is the parameter. Let a = 2
so that g
(
x
)
= 2ƒ
(
x
)
. Complete the input-output table and then
graph g
(
x
)
with f(x). In general, how is the graph of
g
(
x
)
= a ⋅ ƒ
(
x
)
related to the graph of ƒ
(
x
)
when a is greater
than 1?
xf(x) 2f(x)
-1 -2 -4
124
3 -2
52
Now try a value of a between 0 and 1 in g
(
x
)
= a ⋅ ƒ
(
x
)
. Let a =
1
_
2
so that g
(
x
)
=
1
_
2
ƒ
(
x
)
.
Complete the input-output table and then graph g
(
x
)
with f(x). In general, how is the graph
of g
(
x
)
= a ⋅ ƒ
(
x
)
related to the graph of ƒ
(
x
)
when a is a number between 0 and 1?
xf(x)
1
__
2
f(x)
-1 -2 -1
121
3 -2
52
Now graph g
(
x
)
= ƒ
(
1
__
b
⋅ x
)
where b is the parameter. Let b = 2 so that g
(
x
)
= ƒ
(
1
__
2
x
)
.
Complete the mapping diagram and then graph g
(
x
)
with f(x). (To complete the mapping
diagram, you need to find the inputs for g that produce the inputs for f after you multiply
by
1
_
2
. Work backward from the inputs for f to the missing inputs for gby multiplying by 2.)
In general, how is the graph of g
(
x
)
= ƒ
(
1
__
b
x
)
related to the graph of ƒ
(
x
)
when b is a number
greater than 1?
Make a Conjecture How would you expect the graph of g
(
x
)
= ƒ
(
1
__
b
⋅ x
)
to be related
to the graph of ƒ
(
x
)
when b is a number between 0 and 1?
?
?
?
?
6
10
Input
for g
Input
for f
Output
for f
Output
for g
-
2
2
-
1
1
3
5
-
2
2
-
2
2
-
2
2
-
2
2
·
1
2
?
?
-
4
-
6
4
6
2
0
462
x
-
2
-
4
-
6
-4
4
For a > 1, the graph of g(x) = a ⋅ f(x) is the
graph of f(x) stretched vertically (away
from the x-axis) by a factor of a.
See graph in margin.
See graph in margin.
See margin.
-1
1
For 0 < a < 1, the graph of g(x) = a ⋅ f(x) is
the graph of f(x) compressed vertically
(toward the x-axis) by a factor of a.
For b > 1, the graph of g(x) = f
(
1
__
b
⋅ x
)
is
the graph of f(x) stretched horizontally
(away from the y-axis) by a factor of b.
Module 1
25
Lesson 3
Explore 2
Investigating Stretches and
Compressions of Function Graphs
Questioning Strategies
On the coordinate plane, what is the difference
between a vertical stretch and a vertical
compression? In a vertical stretch, the points of the
graph are pulled away from the x-axis. In a vertical
compression, they are pulled toward the x-axis.
On the coordinate plane, what is the difference
between a horizontal stretch and a vertical
stretch? In a horizontal stretch, the points of the
graph are pulled away from the y-axis. In a vertical
stretch, the points of the graph are pulled away
from the x-axis.
Integrate Technology
A graphing calculator can be used to explore
the effects of different values of b in the
function g
(
x
)
= f
(
1
_
b
x
)
on the graph of f
(
x
)
. Choose a
simple function for f
(
x
)
, graph the function, and have
students suggest different values for b. Graph the
transformed functions, and have students compare
the graphs to see how changing the parameter affects
the graph.
Integrate Math Processes and Practices
Focus on Abstract and Quantitative
Reasoning
MPP2 Prompt students to recognize that when the
graph of a function passes through the origin, a
transformation involving a stretch or a compression of
the function will not affect the point at the origin. Ask
students to justify how this is possible, when all points
on either side of the origin are affected.
Answers
-
4
-
6
4
6
2
y
0
462
x
-
2
-
4
-
6
B.
-
4
4
2
y
0
468102
x
-
2
-
4
-
6
C.
D. For 0 < b < 1, the graph of g(x) = f
(
1
__
b
⋅ x
)
is the graph of f(x) compressed horizontally
(toward the y-axis) by a factor of b.
Lesson 1.3 25
/ 12
End of Document
423

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FAQs

What are the main types of transformations for function graphs?
The main types of transformations for function graphs include translations, stretches, compressions, and reflections. Translations shift the graph horizontally or vertically without altering its shape. Stretches and compressions change the graph's size, either vertically or horizontally, while reflections flip the graph over a specified axis. Understanding these transformations is crucial for analyzing how changes in function equations affect their graphical representations.
How does a vertical translation affect a function graph?
A vertical translation of a function graph occurs when a constant is added or subtracted from the function's output. For example, in the function g(x) = f(x) + k, where k is a positive number, the graph shifts upward by k units. Conversely, if k is negative, the graph shifts downward. This transformation affects the range of the function but does not change its domain.
What is the effect of a horizontal stretch on a function graph?
A horizontal stretch occurs when the input variable x is multiplied by a factor less than 1, such as in g(x) = f(1/b * x), where b > 1. This transformation causes the graph to expand away from the y-axis, making it wider. It alters the x-coordinates of points on the graph while keeping the y-coordinates unchanged, effectively stretching the graph horizontally.
How do reflections across the axes impact function graphs?
Reflections across the axes significantly alter the orientation of function graphs. Reflecting a graph across the x-axis, as in g(x) = -f(x), flips the graph upside down, changing the signs of all output values. Reflecting across the y-axis, as in g(x) = f(-x), mirrors the graph horizontally, altering the signs of the input values. These transformations are essential for understanding symmetry in functions.

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