November 11th copyright2009merrydavidson

Get a graphing calculator.

Put pass in the drawer if you are using mine.

Warm-Up

Simplify:

x2(x4) = 3x4(6x-2) =

(2x3)4=

When multiplying a like base you ADD the exponents. = x6

Move the x-2 to the bottom then subtract exponents. Only move the x not the 6. = 18x2

When raising to a power you multiply the the exponents of the variable but take 2 to the fourth power. = 16x12

Exponential Functions 3.1

Definition:

xy ab , where 0b

Notice the variable is in the exponent.

“b” is the base. “a” is the vertical stretch or compression.

Determine if these are exponential growth or decay.11

( ) (2)3

xf x

1( ) 4 2xf x

( ) .3 2xf x

growth

.3 is decay, Ry makes it growth

.3 is the base not the “a”

2 is the base which is growth but the negative x is reflect over the y which turns growth into decay.

Graph using a graphing calculator. Sketch the following in your notes on the same graph.

1) 2) 3)

Label each graph.

2xy 5xy 10xy

Characteristics of Parent Exponential Functions with b>1

1) Smooth, continuous, increasing curve

2) Domain:

3) Range:

,

0, y = 10xy = 5x

y = 2x

y = 0

Horizontal asymptote

What point do they all have in common?

(0, 1)This is called the pivot point.

The bigger the base, the steeper the graph

What else do you notice?

They all go thru

(1, base)

The graphs approach y = 0 but do not touch it.

Transformations on exponential functions, are like doing transformations on all of the other parent functions.

Remember ALL of the rules for inside of the function/outside of the function.

INSIDE affects the x-coordinate

OUTSIDE affects the y-coordinate

Inside of the function means inside of the exponent.

Use this T-chart for graphing all exponential functions.

x y

0 1

1 base

12xy

Where does this graph move?

2

-1

0

4)

left 1y = 0

Use this T-chart for graphing all exponential functions.

x y

0 1

1 baseWhere does this graph move? 3

3

5

23 xy5)

Up 2

y = 2

The asymptote moves up 2

Use this T-chart for graphing all exponential functions.

x y

0 1

1 base Where does this graph move?

5

0

-1

6)5 1xy

Ry, down 1

0

4

y = -1

You do these 3:

7)

8)

9)

13xy

2 2x

y

62xf x

y = 0

y = -2

y = 0

Write the end behavior in limit notation for the 6 graphs you just did.

6) lim ( ) ; lim ( ) 1x x

f x f x

5) lim ( ) 2; lim ( )x x

f x f x

8) lim ( ) ; lim ( ) 2x x

f x f x

9) lim ( ) 0; lim ( )x x

f x f x

4) lim ( ) 0; lim ( )x x

f x f x

7) lim ( ) 0; lim ( )x x

f x f x

State the y-intercept and pivot point for each of the six graphs.

Remember to find the y-intercept let x = 0.The pivot point is what (0,1) became after

transformations.y-intercept pivot point

4) (0,2) (-1,1)5) (0,3) (0,3)6) (0,0) (0,0)7) (0,-1/3) (1,-1)8) (0,-3) (0,-3)9) (0,1/64) (6,1)

TIME OUT….

“Pi” represents the irrational number that is approximately equal to 3.14

“i” represents the imaginary number which is the square root of -1.

“e” is the Euler Number

Approximately = 2.718

For every value of n: 1(1 )ne

n

You do not have to memorize this. You only need to know that 2.7e

The Natural Base Exponential Function

xy eWe can do transformations with base “e” just like any other base.

x y

0 1

1 base

Where does this graph move?

2.7

2

3

10)

right 2

y = 0

2xy e