Lesson 22
Given below is a table of inputs, outputs, and ordered pairs for the function ( ) = 2 , as well as its graph.

Inputs

Outputs

−2
−1
0
1
2

( ) (−2) = 4 (−1) = 1 (0) = 0 (1) = 1 (2) = 4

Ordered
Pairs
(, ( ))
(−2, 4)
(−1, 1)
(0, 0)
(1, 1)
(2, 4)

I have plotted the ordered pairs above in the graph below.
Function f

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Example 1: Use the function on the previous page, its table, and its graph to answer the following:
a. How can the function ( ) = 2 + 2 be written in terms of the function ?
( ) = ( ) + 2
b. Is the new function changing the inputs or the outputs of the original function ? It is changing the ‘outputs’.
c. Make a table and graph the function . (Keep the same input values as function f on previous page.)
Plot the points for function g from the table. The graph will be ‘shifted up’ 2 units from the graph of f g(x) as seen below.

Inputs

-2
-1
0
1
2

Outputs
( )
6
3
2
3
6

How has function f been transformed to create function g?
It has been shifted vertically upward 2 units.

d. How can the function ℎ( ) = 2 − 2 be written in terms of the function ? ℎ( ) = ( ) − 2
e. Is the new function ℎ changing the inputs or the outputs of the original function ? It is changing the ‘outputs’.
f. Make a table and graph the function ℎ. (Keep the same input values.)
Plot these points on the graph at the right. The new graph will be
‘shifted down’ 2 units from the original graph of f shown. Inputs

-2
-1
0
1
2

Outputs
ℎ( )
2
-1
-2
-1
2

How has function f been transformed to create function h? It has been shifted vertically downward 2 units.

2

Example 2: Use the original function , its table and graph (on page 1 and below) to answer the following:
a. write the function ( ) = ( + 2)2 in terms of the function ?
( ) = ( + 2)
b. is the new function changing the inputs or the outputs of the original function ? It is changing the ‘inputs’.
c. make a table and graph the function . (Keep the same ‘output’ values.) Let y’s stay the same and find the x values. f(x) j(x)
-2
4
-1
1
Inputs Outputs
0
0

( )
1
1
-4
4
2
4
-3
1
±3
9
-2
0
-1
1
0
4

Plot the points for function j from the table above. Function j is function f shifted how? The ‘new’ graph has been shifted horizontally 2 units left.

(This problem continued on the next page.)

3

d. write the function ( ) = ( − 2)2 in terms of the function ?
() = ( − 2)

e. is the new function changing the inputs or the outputs of the original function ? It is changing the ‘inputs’; each x is 2 less than original x.
f. make a table and graph the function . (Keep the same ‘output’ values.) ()
Inputs Outputs f(x) ( )
-2 4
0
4
-1 1
1
1
0 0
2
0
1 1
3
1
2 4
4
4
How has function f been transformed to create function k? It has been horizontally shifted right 2 units.

Plot the points for function k as listed in the table above middle.

4

The graph of ( ) = ( ) + 2 is the graph of shifted upward units, which means 2 is added to the outputs of .
The graph of ℎ() = ( ) − 2 is the graph of shifted downward units, which means 2 is subtracted from the outputs of .
The graph of () = ( + 2) is the graph of shifted left units, which means 2 is subtracted from the inputs of .
The graph of () = ( − 2) is the graph of shifted right units, which means 2 is added to the inputs of .
***When the change takes place outside the parentheses, do exactly what you see to the outputs. When the change takes place inside the parentheses, do the opposite of what you see to the inputs.***
This will be true regardless of the operation; if is occurring outside the parentheses, do what you see, if it is occurring inside the parentheses, do the opposite.

5

Given below is a table of inputs, outputs, and ordered pairs for some function , as well as its graph. f (x)
Ordered
Inputs
Outputs
Pairs

( )
(, ( ))
−8
(−8) = 0 (−8, 0)
−4
(−4) = 4 (−4, 4)
(0, 0)
0
(0) = 0
4
(4) = −4 (4, −4)
(8, 0)
8
(8) = 0
Write the ordered pairs for the major points shown below.
Function f

(-4, 4)

(0, 0)

(-8, 0)

(8, 0)

(4, -4)

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Example 3: Use the previous table and graph of the function (graph is also below) to answer the following:
a. Is the function ( ) = 2( ) changing the inputs or outputs of the function ? It is changing the outputs; they are multiplied by 2.
b. Make a table and graph the function . Begin with the same ‘inputs’
(x values) used in the table for function f.
Plot the points for function m below. The graph of f is shown.

m(x) x f(x)

-8
-4
0
4
8

0
4
0
-4
0

Inputs

-8
-4
0
4
8

Outputs
( )
0
8
0
-8
0

How has function f been transformed to create function m? It has been stretched vertically by a factor of 2.

1

c. is the function ( ) = 2 ( ) changing the inputs or outputs of the function ? It is changing the outputs; they are half the original.
d. make a table and graph the function . Begin with the same input values as for function f.
Plot the points for function n x -8
-4
0
4
8

f(x)
0
4
0
-4
0

Inputs

-8
-4
0
4
8

below. The graph of f is shown.

Outputs
( )
0
2
0
-2
0

How has function f been transformed to create function n? It has been compressed vertically by a factor ½.
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Example 4: Use the table and graph of the function to answer the following: a. Is the function ( ) = (2 ) changing the inputs or outputs of the function ? The ‘inputs’ are changed. Each x is half the original.
b. Make a table and graph the function . (Keep the same ‘outputs’ as function f.)
Plot the point for function p p(x) x f(x)
-8
0
-4
4
0
0
4
-4
8
0
-10 -4
10
4

Inputs

-4
-2
0
2
4
-5
5

below. Function f is already graphed. Outputs
( )
0
4
0
-4
0
-4
4

How does function p compare to function f ? How has it been ‘transformed’? It has been compressed horizontally by a factor of ½.
1

c. Is the function () = ( ) changing the inputs or outputs of the function ? The
2
‘inputs’ are changed; each is double the original.
d. Make a table and graph the function . (Keep the same ‘outputs’ as function f.)

q(x) x f(x)
-8
0
-4
4
0
0
4
-4
8
0
-10 -4
10
4

Inputs

-16
-8
0
8
16
20

Outputs
( )
0
4
0
-4
0
4

How has function f been transformed to make function q? It has been stretched horizontally by factor of 2.

8

The graph of () = 2 ( ) is the graph of stretched vertically, which means the outputs of are multiplied by .
1

The graph of ( ) = 2 ( ) is the graph of compressed vertically,

which means the outputs of are multiplied by (divided by 2 ).
The graph of ( ) = (2 ) is the graph of compressed horizontally, which means the inputs of are divided by .
1

The graph of ( ) = (2 ) is the graph of stretched horizontally,

which means the inputs of are divided by (multiplied by 2).
When the change takes place outside the parentheses, do exactly what you see to the outputs. When the change takes place inside the parentheses, do the opposite of what you see to the inputs.

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Given below is a table of inputs, outputs, and ordered pairs for the function ( ) = √ , as well as its graph.
Inputs

Outputs

0
1
4
9

( )
(0) = 0
(1) = 1
(4) = 2
(9) = 3

Ordered
Pairs
(, ( ))
(0, 0)
(1, 1)
(4, 2)
(9, 3)

Label the ‘major’ points in the graph of function f below.
Function f

(9, 3)
(4, 2)
(1, 1)

(0, 0)

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Example 5:
() = −()
a. Write the function ( ) = −√ in terms of ( ) = √ .
b. Is the new function changing the inputs or the outputs of the original function ? It is changing the ‘outputs’.
c. Make a table and graph the function . (Keep the same ‘input’ values as function f. f (x) x f (x)
0
0
1
1
4
2
9
3

Inputs

0
1
4
9

Outputs
( )
0
-1
-2
-3

Function g is a vertical reflection of function f.

d. write the function ℎ( ) = √− in terms of ( ). ℎ( ) = (−)
e. is the new function ℎ changing the inputs or the outputs of the original function ? It is changing the ‘inputs’.
f. make a table and graph the function ℎ. Keep the same ‘output’ values as function f . x 0
1
4
9

f (x)
0
1
2
3

Inputs

0
-1
-4
-9

Outputs
ℎ( )
0
1
2
3

Function h is a horizontal reflection of function f.
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The graph of ( ) = −() is the graph of reflected through the horizontal axis (vertical reflection), which means the outputs of are multiplied by −.
The graph of ℎ() = (−) is the graph of reflected through the vertical axis (horizontal reflection), which means the inputs of are divided by −.
Example 6: If the point is on the graph of a function , find the corresponding point on the graphs of the given functions. Be aware of order of operations.
a. (0, 5); = ( + 2) − 1
b. (3, −2); = 2 ( − 3) + 2
b. A
Subtract 2 from the x-coordinate
Add 3 to the x-coordinate (shifted 3 units right).
(shifted left 2 units). Subtract 1 from the y-coordinate (shifted down 1 unit).
Corresponding point: (−2, 4)

c. (3, 9);

1

Multiply the y-coordinate by 2, then add 2
(stretched vertically by factor 2, then shifted up
2 units).
Corresponding point: (6, −2)

1

d. (−2, 1); = −3 (2 ) − 5

= 3 (2 ) − 1

Multiply the x-coordinate by 2 (horizontal stretch of factor 2). Multiply y-coordinate by 1/3, and then subtract 1 (vertical compression of factor 1/3 , then shifted down 1 unit. New Point: (6, 2)

Multiply the x-coordinate by ½ (or divide by 2; horizontal compression). Multiply the ycoordinate by -3 (vertical reflection and vertical stretch), then subtract 5 (shifted down 5 units).
Corresponding point: (−1, −8)

Example 7: Explain how the graphs of the following functions compares to the graph of = (). Be aware of order of operations.
a. = ( − 2) + 3
b. = (− ) − 2
b. a
The new function has been shifted 2 units
The new function has been reflected horizontally right, then shifted up 3 units.

about the y-axis, then shifted down 2 units.

1

1

c. = (2 ) − 3
The new function has been horizontally stretched by a factor of 2, then shifted down
3 units.

d. = −2 (3 )
The new function has been horizontally stretched by a factor of 3, then vertically reflected over the x-axis and stretched vertically by a factor of 2.

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Given below are the graphs of functions with either even or odd degrees on variables; what do you notice about each with regards to symmetry?
() = 2

() =

(-2, 4)
(2, 2)

(2, 4)

(-2, -2)

Symmetry about the origin.

(1, 1)

(-1, -1)

ℎ() = 5

() = 6

13

Symmetry about the y-axis.

() = 4

() = 3

Even function:
- A function is even if an input and its negative result in the same output.
- (− ) = ( ), for every in the domain (Opposite x yields same y.)
- ( ) = 2 is an example of an even function. o (2) = 4 o (−2) = 4
- The graph of an even function is symmetric with respect to the vertical axis.
Odd function:
- A function is odd if an input and its negative result in outputs with opposites signs (same absolute value).
- (− ) = −( ), for every in the domain (Opposite x yields opposite y.)
- ( ) = 3 is an example of an odd function. o (2) = 8 o (−2) = −8
- The graph of an odd function is symmetric with respect to the origin. This means the origin is the midpoint of the segment connecting points (x, y) and
(-x, -y).
Example 9: Determine whether the following polynomial functions are even, odd or neither.
a. () = 2 − 3 + 4

(−) = (−)2 − 3(−) + 4 = 2 + 3 + 4

x 2  3x  4 and x 2  3x  4 are neither the same or opposites.
This function is neither even nor odd.

b. () = 3 3 +

(−) = 3(−)3 + (−) = −3 3 −
3x3  x and  3x3  x are opposites. This function is odd.
c. ℎ() = 4 4 − 2 + 5 ℎ(−) = 4(−)4 − (−)2 + 5 = 4 4 − 2 + 5
The outputs are both 4 x4  x2  5 . This function is even.

If a function is not a polynomial, there are two ways to determine whether it is even, odd or neither:
1. algebraically by substituting − for as above examples a, b, and c.
a. if (− ) = ( ), the function is even
b. if (− ) = −( ), the function is odd
OR
2. geometrically by checking for symmetry
(continued on the next page)
14

a. If the graph of the function is symmetric about the vertical axis, the function is even.
b. If the graph of the function is symmetric about the origin, the function is odd.
Example 10: Determine whether the following functions are even, odd or
b. There is no neither. a. There is symmetry about symmetry about the the y-axis or the
a. ( ) = | |
b. ( ) = √ y-axis. The function origin. The is even.

function is neither even nor odd. b. ( ) = √

3

c. ℎ( ) = √ 3 −

The origin would be the midpoint of corresponding points; symmetry about the origin.
The function is odd.

d. ( ) = −√9 − 2

There is symmetry about the y-axis.
This function is even.

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