8-2 Properties of Exponential Functions

The function f(x) = bx is the parent of a family of exponential functions for each value of b. The factor a in y = abx stretches, shrinks, and/or reflects the parent.

Summary Families of Exponential Functions

Parent functions (b > 0, b = 1): y = bx

Stretch (|a| > 1)

Shrink (0 < |a| < 1)

Reflection (a < 0) in x-axis

Translation (horizontal by h; vertical by k): y = bx-h + k

Combined: y = abx-h + k

y = abx}

example 1: Graphing y = abx for 0<a<1

Graph y = and y = . Label the asymptote of each graph.

Step 1 Make a table Step 2 Graph the function

x12

2 x2

2

1

x

-2

-1

0

1

2

3

xy 22

1 xy 2

2

1

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9 10x

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

1

2

3

4

5

6

7

8

9

10

y

x

The y-intercept is a, or ½

The asymptote is y = 0 for both graphs

The y-intercept is a, or ½

example 2: Translating y = abx

Graph the stretch y = 8(½)x and then the translation y = 8(½)x + 2 + 3.

Step 1 Graph y = 8(½)x. The horizontal

asymptote is y = 0.

Step 2 For y = 8(½)x + 2 + 3, h = -2 and

k = 3. So shift the y = 8(½)x

graph 2 units left and 3 units up.

The horizontal asymptote is y = 3.-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9 10

x

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

1

2

3

4

5

6

7

8

9

10

y

x

Class Work 8-2

Graph each function.

1. y = -4(2)x 2. y = -3x

3. Graph the stretch y = 9(3)x

4. y = 9(3)x+1

5. y = 9(3)x-3 – 1