Exponential Functions and Their Graphs

2

Exponential Function Families

• We’ve already learned about – This is the parent function

• We’ll expand this to – Note a=1 and h=0 in the parent function

xaby

kaby hx )(

3

Component Property

+ k Vertical shift up- k Vertical shift down- h Horizontal shift to the right+ h Horizontal shift to the left

a > 1 Stretcha < 1 Compression

-a Reflection

kaby hx )(

4

The graph of f(x) = ax, a > 1y

x(0, 1)

Domain: (–, )

Range: (0, )

Horizontal Asymptote y = 0

4

4

5

The graph of f(x) = ax, 0 < a < 1y

x(0, 1)

Domain: (–, )

Range: (0, )Horizontal Asymptote

y = 0

4

4

6

Example: Sketch the graph of f(x) = 2x.

x

x f(x) (x, f(x))-2 ¼ (-2, ¼)-1 ½ (-1, ½)0 1 (0, 1)1 2 (1, 2)2 4 (2, 4)

y

2–2

2

4

7

Example: Sketch the graph of g(x) = 2x – 1. State the domain and range.

x

yThe graph of this function is a vertical translation of the graph of f(x) = 2x

down one unit .

f(x) = 2x

y = –1 Domain: (–, )

Range: (–1, )

2

4

8

Example: Sketch the graph of g(x) = 2-x. State the domain and range.

x

yThe graph of this function is a reflection the graph of f(x) = 2x in the y-axis.

f(x) = 2x

Domain: (–, )

Range: (0, ) 2–2

4

9

The graph of f(x) = ex

y

x2 –2

2

4

6

x f(x)-2 0.14-1 0.380 11 2.722 7.39

10

The irrational number e, where

e 2.718281828…

is used in applications involving growth and decay.

11

Pert

12

Continuously Compounded Interest

rtePtA )(

13

Pert ExampleSuppose you won a contest at the start of 5th grade that deposited $3000 in an account that pays 5% annual interest compounded continuously. How much will you have in the account when you enter high school 4 years later? Express the answer to the nearest dollar.

rtePtA )()4)(05(.3000)4( eA

2.3000)4( eA 3664$)4( A