Warm Up

Complete the Grok Activity on the back of your homework (the one with people at the top)

Exponential Functions

October 15th

Exponential FunctionsAn exponential function is a

function with a variable in the exponent.

f(x) = a(b)x

Exponential FunctionsParent 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

Let’s review …

Original Function: f(x) = bx

f(x) = -bx

Negatives in front cause a reflection across the x-axis. (make the graph flip)

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:

1. f(x) = 3x – 82. f(x) = -3(2)x

3. f(x) = 4x+5

4. f(x) = 2x + 25. f(x) = 5x-2

f(x) = 3(2)x

X Y-2-10123

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

X Y-2-10123

Domain:

Range:

Parent Graph:

Transformation:

f(x) = -8(.5)x

X Y-2-10123

Domain:

Range:

Parent Graph:

Transformation:

Warm UpComplete the two examples

from the notes yesterday that we did not complete:

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

f(x) = 3x + 2

X Y-2-10123

Domain:

Range:

Parent Graph:

Transformation:

f(x) = -2x - 3

X Y-2-10123

Domain:

Range:

Parent Graph:

Transformation:

Growth and Decay and

InterestOctober 16th

Exponential GrowthExponential 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. 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 .

Example:2. The amount of money spent at the West Outlet Mall in Midtown continues to increase. The total t(x) in millions of dollars can be estimated by the function , 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?

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?

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.

Exponential Growth/Decayy = a(b) x

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.

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?

You Try 2. 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.

Exponential Growth/Decay y = a(b) x

Equation: A = P(1 ± r)t.A represents the final amount.P represents the initial amount.r represents the rate of change expressed as a

decimalt 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.

Initial Amount you Borrow or DepositAnnual Rate of Interest (as a decimal)# of times the interest is compounded per year# of years the amount is deposited or borrowed for

What does n equal when you compound … ?

12

21 52

4

3. The Lieberman’s have $12,000 in a savings account. The bank pays 3.5% interest on savings accounts, compounded monthly. Find the balance in 3 years.

4. Determine the amount of an investment if $300 is invested, at an interest rate of 6.75%, compounded semiannually for 20 years.

Homework Worksheet