2.7
Graphing Techniques
Stretching and Shrinking
Reflecting
Symmetry
Even and Odd Functions
Translations
2.7 - 1
Example 1
STRETCHING OR SHRINKING
A GRAPH
Graph the function
a. g ( x ) = 2 x
Solution Comparing the
table of values for ƒ(x) =
|x| and g(x) = 2|x|, we see
that for corresponding xvalues, the y-values of g
are each twice those of ƒ.
So the graph of g(x) is
narrower than that of ƒ(x).
x
–2
–1
0
1
2
ƒ(x) g(x)
|x| 2|x|
2
4
1
2
0
0
1
2
2
4
2.7 - 2
STRETCHING OR SHRINKING
A GRAPH
Example 1
Graph the function
a. g ( x ) = 2 x
y
x
4
3
–4 –3 –2
x
1
–2
–3
2
3
4
–2
–1
0
1
2
ƒ(x) g(x)
|x| 2|x|
2
4
1
2
0
0
1
2
2
4
–4
2.7 - 3
Example 1
STRETCHING OR SHRINKING
A GRAPH
1
b. h( x ) = x
2
Solution The graph of
h(x) is also the same
general shape as that of
ƒ(x), but here the
coefficient ½ causes the
graph of h(x) to be wider
than the graph of ƒ(x), as
we see by comparing the
tables of values.
x
ƒ(x)
x
h(x)
1
2 x
–2
–1
0
1
2
2
1
0
1
2
1
½
0
½
1
2.7 - 4
STRETCHING OR SHRINKING
A GRAPH
Example 1
1
b. h( x ) = x
2
y
4
3
–4 –3 –2
x
1
–2
2
3
4
x
ƒ(x)
x
h(x)
1
2 x
–2
–1
0
1
2
2
1
0
1
2
1
½
0
½
1
–3
–4
2.7 - 5
Example 1
STRETCHING OR SHRINKING
A GRAPH
c. k ( x ) = 2 x
Solution Use
Property 2 of absolute
value ( ab = a  b )
to rewrite 2 x .
k (=
x ) 2=
x 2 =
x 2x
Property 2
The graph of k(x) = 2 is
the same as the graph of
k(x) = 2xin part a.
2.7 - 6
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 = ƒ(x), then the point (x, ay)
lies on the graph of y = aƒ(x).
a. If a > 1, then the graph of y = aƒ(x) is a
vertical stretching of the graph of y =
ƒ(x).
b. If 0 < a <1, then the graph of y = aƒ(x) is
a vertical shrinking of the graph of
y = ƒ(x)
2.7 - 7
Reflecting
Forming a mirror image of a graph across a
line is called reflecting the graph across
the line.
2.7 - 8
Example 2
REFLECTING A GRAPH ACROSS
AN AXIS
Graph the function.
a. g ( x ) = − x
Solution Every yvalue of the graph
is the negative of
the corresponding
y-value of ƒ( x ) =
x.
x
0
1
4
ƒ(x)
x
0
1
2
g(x)
− x
0
–1
–2
2.7 - 9
Example 2
REFLECTING A GRAPH ACROSS
AN AXIS
Graph the function.
a. g ( x ) = − x
y
4
x
0
1
4
ƒ(x)
x
0
1
2
g(x)
− x
0
–1
–2
3
2
–4 –3 –2
1
x
2
3
4
–2
–3
–4
2.7 - 10
Example 2
REFLECTING A GRAPH ACROSS
AN AXIS
Graph the function.
b. h( x )= − x
Solution If we choose
x-values for h(x) that
are the negatives of
those we use for ƒ(x),
the corresponding yvalues are the same.
The graph h is a
reflection of the graph
ƒ across the y-axis.
x
–4
–1
0
1
4
h(x)
ƒ(x)
x
−x
undefined
2
undefined
1
0
0
1
undefined
2
undefined
2.7 - 11
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)
ƒ(x)
x
−x
undefined
2
undefined
1
0
0
1
undefined
2
undefined
–3
–4
2.7 - 12
Reflecting Across an Axis
The graph of y = –ƒ(x) is the same as the
graph of y = ƒ(x) reflected across the x-axis.
(If a point (x, y) lies on the graph of y = ƒ(x),
then (x, – y) lies on this reflection.
The graph of y = ƒ(– x) is the same as the
graph of y = ƒ(x) reflected across the y-axis.
(If a point (x, y) lies on the graph of y = ƒ(x),
then (– x, y) lies on this reflection.)
2.7 - 13
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.
2.7 - 14
Example 3
TESTING FOR SYMMETRY WITH
RESPECT TO AN AXIS
Test for symmetry with respect to the x-axis
and the y-axis.
y x2 + 4
a. =
Solution Replace x
with –x.
y
=
y x +4
2
y = ( −x ) + 4
4
2
=
y x +4
2
Equivalent
–2
x
2
2.7 - 15
TESTING FOR SYMMETRY WITH
RESPECT TO AN AXIS
Example 3
Test for symmetry with respect to the x-axis
and the y-axis.
y x2 + 4
a. =
Solution The result is the same as the
original equation and the graph is symmetric
with respect to the y-axis.
2
=
y x +4
y = ( −x ) + 4
2
Equivalent
=
y x +4
2
2.7 - 16
TESTING FOR SYMMETRY WITH
RESPECT TO AN AXIS
Example 3
Test for symmetry with respect to the x-axis
and the y-axis.
b. =
x y2 − 3
Solution Replace y with –y.
=
x y −3
2
x = ( −y ) − 3
2
Equivalent
= y −3
2
2.7 - 17
Example 3
TESTING FOR SYMMETRY WITH
RESPECT TO AN AXIS
Test for symmetry with respect to the x-axis
y
and the y-axis.
b. =
x y2 − 3
Solution The graph is
symmetric with respect to the
x-axis.
2
=
x y −3
–2
x = ( −y ) − 3
2
Equivalent
4
x
2
= y −3
2
2.7 - 18
Example 3
TESTING FOR SYMMETRY WITH
RESPECT TO AN AXIS
Test for symmetry with respect to the x-axis
and the y-axis.
c. x 2 + y 2 =
16
Solution Substituting –x for x and –y in for y,
we get (–x)2 + y2 = 16 and x2 + (–y)2 = 16.
Both simplify to x2 + y2 = 16. The graph, a
circle of radius 4 centered at the origin, is
symmetric with respect to both axes.
2.7 - 19
Example 3
TESTING FOR SYMMETRY WITH
RESPECT TO AN AXIS
Test for symmetry with respect to the x-axis
and the y-axis.
4
d. 2 x + y =
Solution In 2x + y = 4, replace x with –x
to get –2x + y = 4. Replace y with –y to
get 2x –y = 4. Neither case produces an
equivalent equation, so this graph is not
symmetric with respect to either axis.
2.7 - 20
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.
2.7 - 21
Example 4
TESTING FOR SYMMETRY WITH
RESPECT TO THE ORIGIN
Is this graph symmetric with respect to the
origin?
a. x 2 + y 2 =
16
Solution Replace x with –x and y with –y.
2
2
x +y =
16
Use
parentheses
around – x
and – y
( − x ) + ( − y ) = 16
2
2
Equivalent
x +y =
16
The graph is symmetric with respect to the
origin.
2
2
2.7 - 22
Example 4
TESTING FOR SYMMETRY WITH
RESPECT TO THE ORIGIN
Is this graph symmetric with respect to the
origin?
b. y = x 3
Solution Replace x with –x and y with –y.
y=x
The graph is
symmetric
with respect
to the origin.
3
−y = ( − x )
− y =− x
3
3
Equivalent
y = x3
2.7 - 23
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
2.7 - 24
Even and Odd Functions
A function ƒ is called an even function if
ƒ(–x) = ƒ(x) for all x in the domain of ƒ.
(Its graph is symmetric with respect to the
y-axis.)
A function ƒ is called an odd function is
ƒ(–x) = –ƒ(x) for all x in the domain of ƒ.
(Its graph is symmetric with respect to the
origin.)
2.7 - 25
Example 5
DETERMINING WHETHER FUNCTIONS
ARE EVEN, ODD, OR NEITHER
Decide whether each function defined is
even, odd, or neither.
a. ƒ( x ) = 8 x 4 − 3 x 2
Solution Replacing x in ƒ(x) = 8x4 – 3x2 with
–x gives:
ƒ( − x ) = 8( − x ) − 3( − x ) = 8 x − 3 x = ƒ( x )
4
2
4
2
Since ƒ(–x) = ƒ(x) for each x in the domain of
the function, ƒ is even.
2.7 - 26
Example 5
DETERMINING WHETHER FUNCTIONS
ARE EVEN, ODD, OR NEITHER
Decide whether each function defined is
even, odd, or neither.
b. ƒ( x ) = 6 x 3 − 9 x
Solution
= −=
6 x 3 + 9 x −ƒ( x )
The function ƒ is odd because ƒ(–x) = –ƒ(x).
2.7 - 27
Example 5
DETERMINING WHETHER FUNCTIONS
ARE EVEN, ODD, OR NEITHER
Decide whether each function defined is
even, odd, or neither.
c. ƒ( x ) = 3 x 2 + 5 x
Solution
ƒ( x ) = 3 x 2 + 5
ƒ ( −x =
) 3 ( −x ) + 5 ( −x )
2
Replace x with –x
= 3x − 5x
2
Since ƒ(–x) ≠ ƒ(x) and ƒ(–x) ≠ –ƒ(x), ƒ is
neither even nor odd.
2.7 - 28
Example 6
TRANSLATING A GRAPH
VERTICALLY
Graph g(x) = x– 4
Solution
x
–4
–1
0
1
4
ƒ(x)
x
4
1
0
1
4
y
g(x)
x −4
0
–3
–4
–3
0
ƒ( x ) =
x
x
–4
4
g ( x=
) x −4
(0, – 4)
2.7 - 29
Vertical Translations
2.7 - 30
TRANSLATING A GRAPH
HORIZONTALLY
Example 7
Graph g(x) = x – 4
Solution
x
–2
0
2
4
6
ƒ(x)
x
2
0
2
4
6
y
g(x)
x−4
6
4
2
0
2
ƒ( x ) =
x
x
–4
4
8
g ( x=
) x−4
(0, – 4)
2.7 - 31
Horizontal Translations
2.7 - 32
Horizontal Translations
If a function g is defined by g(x)= ƒ(x – c),
where c is a real number, then for every
point (x, y) on the graph of ƒ, 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 ƒ, 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 ƒ.
2.7 - 33
Example 8
USING MORE THAN ONE
TRNASFORMATION ON GRAPHS
Graph the function.
a. ƒ( x ) =
− x + 3 +1
Solution To graph this, the lowest point on the
graph of y = xis translated 3 units to the left and
up one unit. The graph opens down because of
the negative sign in front of the absolute value
expression, making the lowest point now the
highest point on the graph. The graph is
symmetric with respect to the line x = –3.
2.7 - 34
Example 8
USING MORE THAN ONE
TRNASFORMATION ON GRAPHS
Graph the function.
a. ƒ( x ) =
− x + 3 +1
y
ƒ( x ) =
− x + 3 +1
3
x
–6
–3
–2
2.7 - 35
Example 8
USING MORE THAN ONE
TRNASFORMATION ON GRAPHS
Graph the function.
b. h( =
x ) 2x − 4
Solution To determine the horizontal translation,
factor out 2.
h( =
x ) 2x − 4
= 2 ( x − 2)
= 2 x − 2
= 2 x −2
Factor out 2.
ab = a  b
2 =2
2.7 - 36
Example 8
USING MORE THAN ONE
TRNASFORMATION ON GRAPHS
Graph the function.
h( x ) = 2 x − 4 = 2 x − 2
b. h( x ) =
− 2x − 4
y
Solution Factor out 2.
4
y= x
2
x
2
4
2.7 - 37
Example 8
USING MORE THAN ONE
TRNASFORMATION ON GRAPHS
Graph the function.
1
2
c. g ( x ) =
− x +4
2
Solution It will have the same shape as that of
y = x2, but is wider (that is, shrunken vertically)
and reflected across the x-axis because of the
negative coefficient and then translated 4 units
up.
2.7 - 38
Example 8
USING MORE THAN ONE
TRNASFORMATION ON GRAPHS
Graph the function.
1
2
c. g ( x ) =
− x +4
2
y
Solution
4
x
–3
3
–3
2.7 - 39
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 = ƒ(x):
1. The graph of y = ƒ(x) + k is translated k units up.
2. The graph of y = ƒ(x) – k is translated k units down.
3. The graph of y = ƒ(x + h) is translated h units to the left.
4. The graph of y = ƒ(x – h) is translated h units to the right.
5. The graph of y = aƒ(x) is a vertical stretching of the
graph of y = ƒ(x) if a > 1. It is a vertical shrinking if 0 < a
< 1.
6. The graph of y = aƒ(x) is a horizontal stretching of the
graph of y = ƒ(x) if 0 < a < 1. It is a horizontal shrinking if
a > 1.
7. The graph of y = – ƒ(x) is reflected across the x-axis.
8. The graph of y = ƒ(– x) is reflected across the y-axis.
2.7 - 40