Warm upA rabbit population starts with 3 rabbits and doubles every month.

1. What is the number of rabbits after 6 months?

Solution

• After 6 months: 192 rabbits

Exponential Functions

Have you ever seen an exponential?

• Have you noticed if you leave food out it might look fine for a few days, then get a little mold, then suddenly be extremely moldy?

OR• Have you notices it takes hot coa coa a long

time to cool enough to drink, but then it gets cold fast?

• These are examples of exponential growth and decay.

Definition of a exponential function

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

• It is used to model growth and decay.• The general form is

Look at warm up to determine what the variables mean

Let’s determine how many rabbits there are in the first 3 months. Month 0 is the starting amount.

As we can see:a= starting numberb= rate of changex= number of time intervals that have passed.

Month Number of rabbits

0 3

1 3 2

2 3 2

3 3 2 2

Example 1

• How would we write this with exponents?

Ask yourself 2 questions: 1. What is being repeated? 2. How many times is it repeated? Answers: 3 is being repeated 5 times.

This equals

Example 2 -You try!

• Rewrite each expression with exponents

1.

2.

Answers

1.

2.

Example 3

• A house was purchased for $120,000 and is expected to increase in value at a rate of 6% per year.

• Write an exponential function modeling the situation.

• What is the value of the house after 3 years?

Example 3: Solution

• A house was purchased for $120,000 and is expected to increase in value at a rate of 6% per year.

• Starting value is 120,000=a• Rate of increase is 1.06=b• Increases per year, so x will represent years.

𝑦=120,000 (1.06 )𝑥

Solution cont…

• How do we find the value after 3 years?• We know x represents years, so plug in

3 for x.

• y= 142921.92

Looking at the “b” in another way:Decay: if b is less than 1

Growth: If b is greater than 1

a = initial amount before measuring growth/decayr = growth/decay rate (often a percent)x = number of time intervals that have passed

Example 4- You try!

• A population of 10,000 bugs increases by 3% every month.

• How many bugs will there be after 5 months?

Solution

• A population of 10,000 bugs increases by 3% every month. • How many bugs will there be after 5 months?

• a=10,000• b= 1+.03 = 1.03• x=5

• y= 11592 bugs

Example 5

• Sarah buys a new car for $18,000. The car depreciates at a rate of 7% per year. How much will the car be worth after 5 years?

Solution

• Sarah buys a new car for $18,000. The car depreciates at a rate of 7% per year. How much will the car be worth after 5 years?

• a=18,000• b= 1-.07 = .93• x=4

• y= 12,522.39

Homework

6.2 Worksheet• Problems:• 1 • 2 • 3• 6