Exponential Graphs

Warm UpSolve:

3 2( ) 3 10f x x x x

Find the Vertex: 2( ) 4 24 5f x x x

2,3

5,0

0253

0103 2

xxx

xxx

xxx

342

24

2

a

bx 31532434 2 y

31,3 V

Definition

In an exponential function, the base is fixed and the exponent is a variable.

xbxf Exponent

base

Exploration

Using your GDC, graph the following exponential functions on the same screen:

xxx yyy 5.1,2,3 321

?INTERSECTalltheydo

POINTwhatAt

.1,0POINTthethrough

passayform

theoffunctionsAllx

Exploration

What do you observe about the function as the base gets larger, and the exponent remains positive?

.

,

,0

grows

functionthefasterthe

basetheLARGERthe

xforthatObserve

.ModelGrowth

acalledisThis

Exploration

Using your GDC, graph the following exponential functions on the same screen:

xxx yyy 5.1,2,3 321

?whyandgraphs

previousthefromdiffer

graphsthesedoHow

.righttheof

insteadlefttheonrise

tofunctiontheforce

ExponentsNegative

.ModelDecay

acalledisThis

Exploration…

Using your GDC, graph the following exponential functions on the same screen:

xx yy

3

1,3 21

?relatedpair

eachofgraphs

theareHow

.axisytheacrossMODELS

GROWTHtheofreflectiona

areMODELSDECAYThe

Continued….

?relatedpairsthe

ofbasestheareHow

xx yy

3

1,3 21

.othereachof

sreciprocalareThey

.33

1: x

x

yassametheisyNOTE

Graph: 2xy

x y

-2 0.25

-1 0.5

0 1

1 2

2 4

HA: y = 0

Domain:

Range:

,

,0

Graph:

2 xy

x y

-2 4

-1 2

0 1

1 0.5

2 0.25

Decreasing!

Domain:

Range:

,

,0

HA: y = 0

Graph:32xy

.

3

functionparentthefrom

leftthetomoved

beenhasgraphThe

Domain:

Range:

,

,0

HA: y = 0

Graph: 2 2xy

.

2

functionparent

thefromupmoved

beenhasgraphThe

HA: y = 2

Domain:

Range:

,

,2

Graph:42 3xy

FunctionParent

3

4

Down

Right

HA: y = -3

Domain:

Range:

,

,3

Graph:12 5xy

FunctionParent

versed

Down

Left

Re

5

1

52 11 xy

Domain:

Range:

,

,5

HA: y = -5

Graph:42 2xy

Domain:

Range:

Parent Function

Right 4Up 2

HA: y = 2

,

,2

Natural exponential function

( ) xf x e

2.718281828...e

Graph:1( ) 3xf x e

Domain:

Range:

Left 1Down 3

,

,3

Logarithmic Function

It’s the inverse of the exponential function

log ya x y a x

Switch the x’s and the y’s!

Graph:2( ) logf x x

Domain:

Range:

Is the inverse of xy 2

Domain:

Range:

,

,0

xy 2

yx 2

,0 ,

Graph:2( ) 3 logf x x

Domain:

Range:

Up 3 from previous example!

,0 ,

Graph:2( ) log ( 4)f x x

Domain:

Range:

Left 4 from Original Example!

,4

,

Graph:2( ) log ( 2)f x x

Domain:

Range:

Right 2 from Original Example!

,2 ,

Graph:2( ) log ( )f x x

Domain:

Range:

Reflected over y-axis.

0, ,

Graph:2( ) logf x x

Domain:

Range:

Reflected over x-axis.

,0 ,

Compound Interest

An infectious disease begins to spread in a small city of population 10,000. After t days, the number of persons who have succumbed to the virus is modeled by the function:

How many infected people are there initially?

How many people are infected after five days?

0.97

10,000( )

5 1245 tv t

e

81250

000,10

12455

000,100

097.0

e

v

1.67812455

000,100 597.0

e

v

Compound Interest

P = Principal

r = rate

t = time in years

n = number of times it’s compounded per year

( ) 1nt

rA t P

n

Compounded: annually n = 1

quarterly n = 4

monthly n = 12

daily n = 365

Find the Final Amount: $8000 at 6.5% compounded quarterly for 8 years

nt

n

rPtA

1

09.134004

065.18000

84

tA

4n

Find the Final Amount: $600 at 9% compounded daily for 20 years

98.3628365

09.01600

20365

tA

365n

Find the Final Amount: $300 at 6% compounded annually for 25 years

tn

n

rPtA

1

56.12871

06.01300

251

tA

1n

Compounded Continuously:

P = Principal

r = rate

t = time in years

E = 2.718281828…

( ) rtA t Pe

Find the Final Amount: $2500 at 4% compounded continuously for 25 years

rtPetA

70.67952500 2504.0 etA

Suppose your are offered a job that lasts one month, and you are to be very well paid. Which of the following methods of payment is more profitable for you? How much will you make?

One million dollars at the end of the month.

Two cents on the first day of the month, 4 cents on the second day, 8 cents on the third day, and, in general, 2n cents on the nth day.

48.474836,21$

21474836482

231

A

centsA

centsA nMore Profitable