Warm up A rabbit population starts with 3 rabbits and doubles every month. 1.What is the number of...
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![Page 1: Warm up A rabbit population starts with 3 rabbits and doubles every month. 1.What is the number of rabbits after 6 months?](https://reader031.fdocuments.us/reader031/viewer/2022032203/56649dba5503460f94aab901/html5/thumbnails/1.jpg)
Warm upA rabbit population starts with 3 rabbits and doubles every month.
1. What is the number of rabbits after 6 months?
![Page 2: Warm up A rabbit population starts with 3 rabbits and doubles every month. 1.What is the number of rabbits after 6 months?](https://reader031.fdocuments.us/reader031/viewer/2022032203/56649dba5503460f94aab901/html5/thumbnails/2.jpg)
Solution
• After 6 months: 192 rabbits
![Page 3: Warm up A rabbit population starts with 3 rabbits and doubles every month. 1.What is the number of rabbits after 6 months?](https://reader031.fdocuments.us/reader031/viewer/2022032203/56649dba5503460f94aab901/html5/thumbnails/3.jpg)
Exponential Functions
![Page 4: Warm up A rabbit population starts with 3 rabbits and doubles every month. 1.What is the number of rabbits after 6 months?](https://reader031.fdocuments.us/reader031/viewer/2022032203/56649dba5503460f94aab901/html5/thumbnails/4.jpg)
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.
![Page 5: Warm up A rabbit population starts with 3 rabbits and doubles every month. 1.What is the number of rabbits after 6 months?](https://reader031.fdocuments.us/reader031/viewer/2022032203/56649dba5503460f94aab901/html5/thumbnails/5.jpg)
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
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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
![Page 7: Warm up A rabbit population starts with 3 rabbits and doubles every month. 1.What is the number of rabbits after 6 months?](https://reader031.fdocuments.us/reader031/viewer/2022032203/56649dba5503460f94aab901/html5/thumbnails/7.jpg)
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
![Page 8: Warm up A rabbit population starts with 3 rabbits and doubles every month. 1.What is the number of rabbits after 6 months?](https://reader031.fdocuments.us/reader031/viewer/2022032203/56649dba5503460f94aab901/html5/thumbnails/8.jpg)
Example 2 -You try!
• Rewrite each expression with exponents
1.
2.
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Answers
1.
2.
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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?
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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
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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
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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
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Example 4- You try!
• A population of 10,000 bugs increases by 3% every month.
• How many bugs will there be after 5 months?
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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
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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?
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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
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Homework
6.2 Worksheet• Problems:• 1 • 2 • 3• 6