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Exponential FunctionsAn is a function

of the form ,

where 0, 0, and 1,

and t

expone

exponent vahe riab

ntial f

must be a .

unction

le

xy

b

b

b

a

a

constant a is the initial value of f(x) at x = 0,b is the base

Let’s examine exponential functions. They are different than any of the other types of functions we’ve studied because the independent variable is in the exponent.

xxf 2

Let’s look at the graph of this function by plotting some points. x 2x

3 8 2 4 1 2 0 1

-1 1/2 -2 1/4 -3 1/8

2 -7 -6 -5 -4 -3 -2 -1 1 5 7 3 0 4 6 8

7

123456

8

-2-3-4-5-6-7

2

121 1 f

Recall what a negative exponent means:

BASE

a> 0, b > 1 exponential growth, 0<b< 1 Exponential Decay

Pg 280

This says that if we have exponential functions in equations and we can write both sides of the equation using the same base, we know the exponents are equal.

If au = av, then u = v

82 43 x The left hand side is 2 to the something. Can we re-write the right hand side as 2 to the something?

343 22 xNow we use the property above. The bases are both 2 so the exponents must be equal.

343 x We did not cancel the 2’s, We just used the property and equated the exponents.

You could solve this for x now.

The Equality Property for Exponential Functions

Let’s try one more:8

14 x The left hand side is 4

to the something but the right hand side can’t be written as 4 to the something (using integer exponents)

We could however re-write both the left and right hand sides as 2 to the something.

32 22 x

32 22 xSo now that each side is written with the same base we know the exponents must be equal.

32 x

2

3x

Check:

8

14 2

3

8

1

4

1

2

3 8

1

4

12 3

Example 1:32x 5 3x 3

(Since the bases are the same wesimply set the exponents equal.)

2x 5 x 3x 5 3

x 8

Here is another example for you to try:Example

1a:23x 1 21

3x 5

Example 2: (Let’s solve it now)

32x 3 27x 1

32x 3 33(x 1) (our bases are now the sameso simply set the exponents equal)2x 3 3(x 1)

2x 3 3x 3

x 3 3

x 6

x 6

Let’s try another one of these.

Example 3

16x 1 1

32

24(x 1) 2 5

4(x 1) 54x 4 5

4x 9

x 9

4

Remember a negative exponent is simply another way of writing a fraction

The bases are now the sameso set the exponents equal.

All of the transformations that you learned apply to all functions, so what would the graph of look like?

xy 232 xy

up 3

xy 21up 1

Reflected over x axis 12 2 xy

down 1right 2

xy 2

Reflected about y-axis This equation could be rewritten in a different form: x

xxy

2

1

2

12

So if the base of our exponential function is between 0 and 1 (which will be a fraction), the graph will be decreasing. It will have the same domain, range, intercepts, and asymptote.

There are many occurrences in nature that can be modeled with an exponential function. To model these we need to learn about a special base.

Slide 3- 14

The Nature of Exponential FunctionsA Table of Values

Determine formulas for the exponential function and whose values are

given in the table below.

g h

1

Because is exponential, ( ) . Because (0) 4, 4.

Because (1) 4 12, the base 3. So, ( ) 4 3 .

x

x

g g x a b g a

g b b g x

1

Because is exponential, ( ) . Because (0) 8, 8.

1Because (1) 8 2, the base 1/ 4. So, ( ) 8 .

4

x

x

h h x a b h a

h b b h x

 

The Base “e” (also called the natural base)

To model things in nature, we’ll need a base that turns out to be between 2 and 3. Your calculator knows this base. Ask your calculator to find e1. You do this by using the ex button (generally you’ll need to hit the 2nd or yellow button first to get it depending on the calculator). After hitting the ex, you then enter the exponent you want (in this case 1) and push = or enter. If you have a scientific calculator that doesn’t graph you may have to enter the 1 before hitting the ex. You should get 2.718281828

Example for TI-83

xxf 2

xxf 3

xexf

To Do

• Complete pg 247- 38, 40, 42, 44 Analyze• Domain, Range, Continuity, Decreasing,

Increasing, Symmetry(even, odd), Bounded, Extrema, Horizontal Asymptotes, Vertical Asymptotes, Using limits describe behavior of the function as x approaches the vertical asymptote, End behavior

• Pg 286 #2, 4, 6, 12, 24, 66• Homework: pg 287 1-19 odds Read Sec. 3.2

Acknowledgement

I wish to thank Shawna Haider from Salt Lake Community College, Utah USA for her hard work in creating this PowerPoint.

www.slcc.edu

Shawna has kindly given permission for this resource to be downloaded from www.mathxtc.com and for it to be modified to suit the Western Australian Mathematics Curriculum.

Stephen CorcoranHead of MathematicsSt Stephen’s School – Carramarwww.ststephens.wa.edu.au