Warm up A rabbit population starts with 3 rabbits and doubles every month. 1.What is the number of rabbits after 6 months?
Dec 24, 2015
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