Slide 1

Vertical Stretching or Shrinking of
the Graph of a Function
Suppose that a > 0. If a point (x, y) lies on
the graph of y = (x), then the point (x, ay)
lies on the graph of y = a(x).
a. If a > 1, then the graph of y = a(x) is a
vertical stretching of the graph of y =
(x).
b. If 0 < a <1, then the graph of y = a(x) is
a vertical shrinking of the graph of
y = (x)
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Slide 2

Horizontal Stretching or Shrinking
of the Graph of a Function
Suppose a > 0. If a point (x, y) lies on the
graph of y = (x), then the point ( x , y) lies
a
on the graph of y = (ax).
a. If 0 < a < 1, then the graph of y = (ax) is
horizontal stretching of the graph of y
= (x).
b. If a > 1, then the graph of y = (ax) is a
horizontal shrinking of the graph of y =
(x).
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Slide 3

Reflecting

Forming a mirror image of a graph across a
line is called reflecting the graph across
the line.

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Slide 4

Example 2

REFLECTING A GRAPH ACROSS
AN AXIS

Graph the function.
a. g ( x )   x

y
4

x
0
1
4

(x)
x
0
1
2

g(x)
 x
0
–1
–2

3

2
–4 –3 –2

1

x
2

3

4

–2
–3
–4

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Slide 5

REFLECTING A GRAPH ACROSS
AN AXIS

Example 2

Graph the function.
b. h( x )   x
y

x

4
3

–4 –3 –2
1
–2

2

3

4

–4
–1
0
x 1
4

h(x)
(x)
x
x
undefined
2
undefined
1
0
0
1
undefined
2
undefined

–3
–4

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Slide 6

Reflecting Across an Axis
The graph of y = –(x) is the same as the
graph of y = (x) reflected across the x-axis.
(If a point (x, y) lies on the graph of y = (x),
then (x, – y) lies on this reflection.
The graph of y = (– x) is the same as the
graph of y = (x) reflected across the y-axis.
(If a point (x, y) lies on the graph of y = (x),
then (– x, y) lies on this reflection.)
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Slide 7

Symmetry with Respect to An
Axis
The graph of an equation is symmetric with
respect to the y-axis if the replacement of x
with –x results in an equivalent equation.
The graph of an equation is symmetric with
respect to the x-axis if the replacement of y
with –y results in an equivalent equation.

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Slide 8

Symmetry with Respect to the
Origin
The graph of an equation is symmetric
with respect to the origin if the
replacement of both x with –x and y
with –y results in an equivalent
equation.

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Slide 9

Important Concepts
1. A graph is symmetric with respect to both xand y-axes is automatically symmetric with
respect to the origin.
2. A graph symmetric with respect to the origin
need not be symmetric with respect to either
axis.
3. Of the three types of symmetry  with respect
to the x-axis, the y-axis, and the origin  a
graph possessing any two must also exhibit the
third type.
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Slide 10

Symmetry with Respect to:
Equation is
unchanged
if:

x-axis

y-axis

Origin

y is replaced
with –y

x is replaced
with –x

x is replaced
with –x and y is
replaced with –
y

y

Example

0

y

y

x

0

x

0

x

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Slide 11

Even and Odd Functions
A function  is called an even function if
(–x) = (x) for all x in the domain of .
(Its graph is symmetric with respect to the
y-axis.)
A function  is called an odd function is
(–x) = –(x) for all x in the domain of .
(Its graph is symmetric with respect to the
origin.)
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Slide 12

Example 5

DETERMINING WHETHER FUNCTIONS
ARE EVEN, ODD, OR NEITHER

Decide whether each function defined is
even, odd, or neither.
a. ( x )  8 x 4  3 x 2
Solution Replacing x in (x) = 8x4 – 3x2 with
–x gives:
(  x )  8(  x )  3(  x )  8 x  3 x  ( x )
4

2

4

2

Since (–x) = (x) for each x in the domain of
the function,  is even.
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Slide 13

Example 5

DETERMINING WHETHER FUNCTIONS
ARE EVEN, ODD, OR NEITHER

Decide whether each function defined is
even, odd, or neither.
b. ( x )  6 x 3  9 x
Solution
 6 x 3  9 x  ( x )

The function  is odd because (–x) = –(x).
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Slide 14

Example 5

DETERMINING WHETHER FUNCTIONS
ARE EVEN, ODD, OR NEITHER

Decide whether each function defined is
even, odd, or neither.
c. ( x )  3 x 2  5 x
Solution
( x )  3 x 2  5
  x   3  x   5  x 
2

Replace x with –x

 3x  5x
2

Since (–x) ≠ (x) and (–x) ≠ –(x),  is
neither even nor odd.
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Slide 15

Vertical Translations

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Slide 16

Horizontal Translations

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Slide 17

Horizontal Translations
If a function g is defined by g(x)= (x – c),
where c is a real number, then for every
point (x, y) on the graph of , there will be a
corresponding point (x + c) on the graph of
g. The graph of g will be the same as the
graph of , but translated c units to the right
if c is positive or c units to the left if c is
negative. The graph is called a horizontal
translation of the graph of .
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Slide 18

Caution Be careful when translating
graphs horizontally. To determine the direction
and magnitude of horizontal translations, find the
value that would cause the expression in
parentheses to equal 0. For example, the graph
of y = (x – 5)2 would be translated 5 units to the
right of y = x2, because x = + 5 would cause x – 5
to equal 0. On the other hand, the graph of
y = (x + 5)2 would be translated 5 units to the left
of y = x2, because x = – 5 would cause x + 5 to
equal 0.
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Slide 19

Summary of Graphing
Techniques
In the descriptions that follow, assume that a > 0, h > 0, and
k > 0. In comparison with the graph of y = (x):
1. The graph of y = (x) + k is translated k units up.
2. The graph of y = (x) – k is translated k units down.
3. The graph of y = (x + h) is translated h units to the left.
4. The graph of y = (x – h) is translated h units to the right.
5. The graph of y = a(x) is a vertical stretching of the
graph of y = (x) if a > 1. It is a vertical shrinking if 0 < a
< 1.
6. The graph of y = a(x) is a horizontal stretching of the
graph of y = (x) if 0 < a < 1. It is a horizontal shrinking if
a > 1.
7. The graph of y = – (x) is reflected across the x-axis.
8. The graph of y = (– x) is reflected across the y-axis.
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