3/7/13 OBJ: SWBAT graph rational functions and recognize

exponential functions.

• Bell Ringer: Start notes for Exponential functions

• Homework Requests: pg 246 #1-29 odds 37, 39, 41, 43

• Homework: p286 #1-19 odds Read Sect. 3.2

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Exponential Functions

An 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 pg 286 #2, 4, 6

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

1 2 3 4 5 6

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

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 21

up 1 Reflected over

x axis 12 2 xy

down 1 right 2

xy 2

Reflected about y-axis This equation could be rewritten in

a different form: x

x

xy

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- 8

The Nature of Exponential Functions A 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

Steps:

𝑓 𝑥 = 𝑎 ∙ 𝑏𝑥

𝑎 = 𝑓 0

𝑏 = 𝑓 1

Ex: pg 287 #12

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

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

1

2 3

Example 1:

32x5

3x 3 (Since the bases are the same we

simply set the exponents equal.) 2x 5 x 3

x 5 3

x 8

Here is another example for you to try:

Example 1a:

23x1

21

3x 5

Example 2: (Let’s solve it now)

32x 3

27x1

32x 3

33(x1) (our bases are now the same

so simply set the exponents equal)

2x 3 3(x1)

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) 5

4x 4 5

4x 9

x 9

4

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

The bases are now the same so set the exponents equal.

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 Corcoran

Head of Mathematics

St Stephen’s School – Carramar

www.ststephens.wa.edu.au