exponential functions and their graphs. rules for exponents exponents give us shortcuts for...

Exponential Functions and Exponential Functions and Their GraphsTheir Graphs

Exponential Functions and Exponential Functions and Their GraphsTheir Graphs

Rules for Exponents• Exponents give us shortcuts for

multiplying and dividing quickly.

• Each of the key rules for exponents has an important parallel in the world of logarithms which is learned later.

Properties of Exponents

Product Rule:

86

2

aa b

bx xx x

xx Quotient

Rule:

Power Rule 23 6ba abx x x x

4 2 6a b a bx x x x x x

Multiplying with Exponents

• To multiply powers of the same base, keep the base and add the exponents.

x y x x x y

x y x y

7 3 5 7 5 3

7 5 3 12 3

Keep x, add exponents 7 +

5

Can’t do anything about the y3 because

it’s not the same base.

Dividing with Exponents

7 5

7 5

7

7

5

5

7 5 7 5

10 12

6 4

10

6

12

4

10 6 12 4 4 8

• To divide powers of the same base, keep the base and subtract the exponents.

Keep 7, subtract 10-

6

Keep 5, subtract 12-

4

Powers with Exponents• To raise a power to a power,

keep the base and multiply the exponents.

This means t7·t7·t7 = t7+7+7

213737 )( ttt x

Copyright © by Houghton Mifflin Company, Inc. All rights reserved 7

The exponential function f with base b is defined by f (x) = abx where a 0, b 0 or 1, and x is any real #.

a is the initial value and b is the base; x is the power to which the base is raised.These are examples of exponential functions:

12 2

12(.25)

t

x

g t

f x

a ,the # outside the ( ) is the y-intercept or

the initial value of the function.

Exponential functions are often asymptotic to the x-axis, meaning

they get closer and closer and closer to it, but can never reach it because

y cannot equal zero.

y =abx

If b > 1, the exponential is increasing, growing as the values of x go up from left to right.


Graph of f(x) = abx, b > 1y

x(a, 1)

Domain: (–, )

Range: (0, )

Horizontal Asymptote y =

0


Graph of f (x) = abx, 0<b <1

y

x

(0, 1)

Domain: (–, )

Range: (0, )

Horizontal Asymptote y = 0

If 0 < b <1, the function is a decreasing exponential.

Determine if the graphs below represent increasing or decreasing functions.

Y=4(1.5)x f(x)=10(1.2)x

Y=6(.42)x F(x)=12(.88)x


What do you notice about the functions:

( ) xf x a( ) xf x a and

They are reflections across the y-axis.

Identify the initial value and the rate of change for each exponential function below. Are they increasing or decreasing?

1. ( ) 12(0.45)

2. ( ) 50(1.23)

33. ( ) 20

5

x

x

x

f x

g x

h x

Transformations of Exponential Functions

Compare the following graph of f(x)=3x and y= 3x+1

Compare the following graph of f(x)=3x and y = 3x-2

f(x)=3x

y = 3x-2

What happened in each case?


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

Domain: (–, ) Range: (–1, )

Y=-1 is the horizontal asymptote

b is the rate of change in the function.b = 1 ± r

where r = the % of change

Nowheresville has a population of 3138. If the population is decreasing at a yearly rate of 3.5%, write an equation to represent this function and determine the population in 5 years.

Y=abx

Mrs. Layton has $5000 to invest in a starting company. If they promise a 6% rate of return each year, what formula

represents this exponential function? In 5 years, how much will her investment be

worth?

Y=abx

Sometimes, rather than an integer, the base of the exponential

function is the irrational number, e.

e is approximately equal to 2.781828… but, because it is irrational, its integers do not repeat in a recognizable pattern.

3 xy e would be an increasing exponential since its base, e, is greater than 1.


Graph of f(x) = ex

y

x2 –2

2

4

6

X-axis is the horizontal asymptote


Graph the following functions:

0.24( ) 2 xf x e0.581( )

2 xg x e

4( ) 3 xf x e2

( ) 2 xf x

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