Exponential Functions

4.3 **You might want

graph paper**

Vocabulary• An exponential function has the form

y = abx, where a ≠ 0 and the base b is a positive number other than 1.

• If a > 0 and b > 1, the function y = abx is an exponential growth function– and b is called the growth factor or

multiplier.

Evaluate Exp. Function

• Example

1.x = 1.4 4. x = 2.x = -3 5. x = 3.x = 1.68 6. x = 3.784

x

xy b2xy

2

Graphing Exp. Functions

• Parent

• Plot the points ( 0, 1) and ( 1, base)• The asymptote will be a horizontal line y = ? and it is always the shift up or down

xy b

Make a table of values.STEP 1

STEP 2 Plot the points from the table.

Graph y = 2x.

Graph the following on your graph paper.

1

4

1. 2

2. 2

3. 2

4. 2 3

5. 2

6. 2 3

x

x

x

x

x

x

y

y

y

y

y

y

Graph the function.

SOLUTION

Plot and (1, 2) .Then,

from left to right, draw a curve that begins just above the x-axis, passes through the two points, and moves up to the right.

0, 12

a.

1a. 42

xy

Graph the function.

b. y = – 52

x

Plot (0, –1) and .

Then,from left to right, draw a curve that begins just below the x-axis, passes through the two points,and moves down to the right.

b. 1, – 52

SOLUTION

Graph

SOLUTION

14 2 3.xy

• Begin by sketching the graph y = 4 ● 2x

• Then translate the graph: y = 4 ● 2x – 1 – 3

•Domain?

•Range?

•Asymptote?

Your TurnGraph each function below. State the

Domain, Range and Asymptote.

• y = 5x

• f(x) = 1/3 ● 4x

• y = 2x + 1 – 1

Your Turn: Answers

Domain: all real #sRange: y > 0Asym.: y = 0

Domain: all real #sRange: y > 0Asym.: y = 0

Domain: all real #sRange: y > -1Asym.: y = -1

Vocabulary

• An exponential decay function has the form y = abx, where a > 0 and 0 < b < 1.

• The base b of an exponential decay function is called the decay factor.

Example 1

Graph y = State the domain and range. Give the equation for the asymptote.

12

x

Example 2• Graph y = -3(1/4)x. State the domain and range.

Give the equation for the asymptote.

Example 3• Graph y = 2(2/3)x – 1 + 1. State the domain and

range. Give the equation for the asymptote.

When b>1, f(x) = bx represents GROWTH

When 0<b<1, f(x) = bx represents DECAY

Growth vs. Decay

Use Functions Involving e• The number denoted by the letter e is

called the natural base e or the Euler number. – Think of e like you do π– e is approximately 2.718281828.

• A function of the form f(x) = aerx is called a natural base exponential function.

• Graph the function. Graph the parent first.

• State the domain and range.a) y = 4e-x + 5

xy e

• Graph the function. State the domain and range.

b) y = e2(x+1) - 2

Exponential Equations

• Exponential equations are equations in which variable expressions occur as exponents.

• Property of Equality for Exponential Equations:If b is a positive number other than 1, then bx = by if and only if x = y.

Examples

1 3 1 11. 3 81 2. 5 25x x x

2 2 4 2 3 13. 9 3 4. 44

x x x

2 235. 8 4 6. 5 25x x x x

21 17. 4 8. 2 16 4

2

xx x x

2 25 43

19. 10.x x x xe e e ee