EXPONENTIAL FUNCTIONSGET A GUIDED NOTES SHEET FROM THE FRONT!

EXPONENTIAL FUNCTIONS

An exponential function is a function with a variable in the exponent.

f(x) = a(b)x

EXPONENTIAL FUNCTIONS

Parent graphs of exponential functions are in the form:

f(x) = bx

Parent function- original function before any changes have been made.

for example: f(x) = 4x

LETS REVIEW THE SHIFTS THAT OCCUR WITH IN EXPONENTIAL FUNCTIONS.

Original Function: f(x) = bx

f(x) = -bx

Negatives in front cause a reflection across the x-axis.

Original Function: f(x) = bx

f(x) = bx-1 f(x) = bx+1

right left

Numbers in the exponents cause horizontal shifts (right, or left).

Original Function: y = bx

f(x) = bx - 1 f(x) = bx + 1 down up

Numbers behind the original function cause vertical shifts (down , and up).

Original Function: f(x) = bx

f(x) = a(b)x

Numbers larger than 1 that are in front of the b value cause a stretch.

IDENTIFY THE PARENT FUNCTION OF EACH, AND THE TRANSFORMATIONS:

PARENT TRANSFORMATIONS

1. f(x) = 3x – 8

2. f(x) = -3(2)x

3. f(x) = 4x+5

4. f(x) = 2x + 2

5. f(x) = 5x-2

F(X) = 3(2)X

X Y-2

-1

0

1

2

3

Domain:

Range:

Parent Graph:

Transformation:

ASYMPTOTES

All exponential functions have horizontal asymptotes.

Notice that the range values of the previous graph were restricted by the horizontal asymptote.

Range is always restricted by the asymptotes.

F(X) = 4X-3

Domain:

Range:

X Y-2

-1

0

1

2

3

Parent Graph:

Transformation:

F(X) = -8(.5)X

Domain:

Range:

Parent Graph:

Transformation:

X Y-2

-1

0

1

2

3

F(X) = 3X + 2

Domain:

Range:

X Y-2

-1

0

1

2

3

Parent Graph:

Transformation:

F(X) = -2X - 3

Domain:

Range:

Parent Graph:

Transformation:

X Y-2

-1

0

1

2

3

EXPONENTIAL GROWTH/DECAYy = a (b) x

Equation: A = P(1 ± r)t.A represents the final amount.P represents the initial amount.r represents the rate of growth/decay

expressed as a decimal.t represents time.

Key words to look for that tell you to use the formula is increase, appreciate and growth.

Key words to look for that tell you to use the formula is decrease, depreciate and decay.

EXPONENTIAL GROWTH

Exponential growth is an initial amount that increases at a steady rate over time.

Exponential growth can be modeled by the function ------------, where a > 0 and b > 1. The base b is the growth factor, which equals 1 plus the percent rate of change expressed as a decimal.

GROWTH GRAPHS

Of the following, which graphs show exponential GROWTH?

EVALUATING AN EXPONENTIAL FUNCTION 1. Example: Suppose 30 flour beetles are left

undisturbed in a warehouse bin. The beetle population doubles each week. The function -------------gives the population after weeks. How many beetles will there be after 56 days? 

Step 1: Convert 56 days to weeks.

Step 2: Evaluate for x = 8.

EXAMPLE: The amount of money spent at West Outlet Mall in

Midtown continues to increase. The total T(x) in millions of dollars can be estimated by the function T(x)=12(1.12)x, where x is the number of years after it opened in 1995.

a) According to the function, find the amount of sales in 2006, 2008 and 2010.

b) Name the y-intercept.

c) What does it represent in this problem?

YOU TRY

3. An initial population of 20 rabbits triples every half year. The function gives the population after x half-year periods. How many rabbits will there be after 3 years?

EXPONENTIAL DECAYExponential Decay occurs when an

initial amount decreases at a steady rate over time.

Exponential decay can be modeled by the function --------- , where a > 0 and b < 1. The base b is the decay factor, which equals 1 minus the percent rate of change expressed as a decimal.

GROWTH GRAPHS

Of the following, which graphs show exponential DECAY?

EXAMPLES1. The original price of a tractor was $45,000. The

value of the tractor decreases at a steady rate of 12% per year.

a. Write an equation to represent the value of the tractor since it was purchased.

b. What is the value of the tractor in 5 years?

EXAMPLES

2. The kilopascal is a unit of measure for atmospheric pressure. The atmospheric pressure at sea level is about 101 kilopascals. For every 1000-m increase in altitude, the pressure decreases about 11.5%. What is the approximate pressure at an altitude of 3000 m?

YOU TRY

3. Find a value of a $20,000 car in five years if it depreciates at a rate of 12% annually. Write the exponential function to model the situation, and find the amount after the specified time.

HOMEWORK

Weekly #6