A function of x is a relation in which no two ordered pairs have the same

x-value.

x f(x)

Does the information given define a function ?

a 1

b 4

c 4

d 3

x f(x) x f(x) x f(x)

a 1

b 2

c 3

d 4

a 5

1 6

2 5

1 -6

2 -5

3 4

a 4

b 5

c 4

b 5

Yes; no two ordered pairs

have the same x-value.

No; two ordered pairs

have the same x-

value.

No; several ordered pairs

have the same x-values.

Yes; no two ordered pairs

have the same x-value.

Math 8H

Heath 12.3Exponential Functions

Heath Algebra 1 McDougal Littell JoAnn Evans

Exponential functions are functions in the form xf x a b

x

x x

x

x x

Here are some examples of exponential f unctions :

1f x 2 f x 3 1 f x

2

2g x 2.2 f x 2 g x 3

3

When a > 1, the graph will be a smooth curve that

approaches the x-axis on the left and rises faster and faster

as you move from left to right.

When 0 < a < 1, the graph will be a smooth curve that falls faster and faster as you move from left to right and

approaches the x-axis on the right.

This graph shows exponential growth.

This graph shows exponential decay.

Graph an exponential function with a > 1.

x -3 -2 -1 0 1 2 3

f(x)

xf x 2

14

2f 2 2

2

1

2

14

0f 0 2

1

12

12

1f 1 2

1

1

2

1

1f 1 2

2 2f 2 2

4 3f 3 2

8

2 4 8

18

3f 3 2

3

1

2

18

Does this graph show exponential growth or

decay?

f(x)

x

x -3 -2 -1 0 1 2 3

f(x)

x -3 -2 -1 0 1 2 3

f(x)

Transformation of an Exponential Function

xf x 2 3

14

12

1 2 4 818

This function shifted the graph of f(x)=2x three units UP.

Look at the table of values for the previous graph f(x) =

2x.

How will the table of values for the new function be

different?

13

41

32

4 5 7 111

38

Each y-value will be 3 greater than in the previous

graph.

f(x)

x xf x 2

Transformation of an Exponential Function

xf x 2 2

This function shifted the graph of

f(x)=2x two units DOWN.

Look at the table of values for the original graph f(x) =

2x.

How will the table of values for the new function be

different?x -3 -2 -1 0 1 2 3

3

14

112

1 0 2 67

18

Each y-value will be 2 less than in the original graph.

f(x)

x xf x 2

x

When a number, b, is added or subtracted to

f x a , it causes a vertical shif t.

x xThe graph of f x 3 5 will shif t the graph of f x 3

5 units UP

x xThe graph of f x 4 3 will shif t the graph of f x 4

3 units DOWN

x x

1 1The graph of f x 1 will shif t the graph of f x

2 2

1 unit UP

Transformation of an Exponential Function

x 1f x 2

This function shifted the graph of f(x)=2x one unit

to the left.

What transformation do you think this graph will show?

x -3 -2 -1 0 1 2 3

12

1 2 4 8 1614

3 1 42 2 16

2 1 1 12 2

2

0 1 12 2 2

2 1 32 2 8

3 1 2 12 2

4

1 1 02 2 1

1 1 22 2 4

f(x)

x

Graph an exponential function with a < 1.

x -3 -2 -1 0 1 2 3

x

1f x

2

2

2

2

1

212

2

2

1

2

4 2

212

112

1

1

1

2

112

14

18

3

3

2

1

312

3

3

1

2

8

Does this graph show exponential growth or

decay?

8 4

012

11

12

12

2

12

14

3

12

18

f(x)

x

Transformation of an Exponential Function

x -3 -2 -1 0 1 2 3

x 2

1f x

2

4

4

2

1

2 212

4

4

1

2

16 8

3

3

2

18

1 212

3

3

1

2

4 2 112

5

5

2

1

3 212

5

5

1

2

32

32 16

0 212

41 2

12

22 2

12

13 2

12

12

This function caused a

shift of two units to the RIGHT.

f(x)

x

x

1f x

2

x

x

To sketch the graph of

f x 2

refl ect the graph of

f x 2

in the x-axis.

xf x 2

xf x 2

One graph is a reflection of another graph in the x-axis if the two graphs are

mirror images of each other.

x

1Now graph f x

2

f(x)

x

Describe the relationship between the graph of function f and the graph of function g.

x x

x x

x x 1

x x

x x 2

x x

f x 3 g x 3 4

f x 5 g x 5 2

f x 2 g x 2

f x 6 g x 6

f x 2.1 g x 2.1

1 1f x g x

2 2

Vertical shift; 4 units UP

Vertical shift; 2 units DOWN

Horizontal shift; 1 unit RIGHTReflection

Horizontal shift; 2 units LEFT

Reflection

Extra Practice:

x 1f x 3

f(x)

x -3 -2 -1 0 1

f(x)

13

1 3 919

x 2f x 1.5

x -1 0 1 2 3 4

f(x) .4 .7 1 1.5.3 2.25