Transcript Intro to Exponential Functions 6.2

Warm up
A 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
𝑦 = 𝑎𝑏
𝑥
Look at warm up to determine what the variables mean
𝑦 = 𝑎𝑏 𝑥
Let’s determine how many rabbits there are in the first 3
months. Month 0 is the starting amount.
Month
Number of rabbits
0
3
1
3∙2=6
2
3 ∙ 2∙ 2 = 12
3
3 ∙ 2 ∙ 2∙ 2 = 24
𝑦 = 3(2)𝑥
As we can see:
a= starting number
b= rate of change
x= number of time intervals that have passed.
Example 1
• How would we write this with exponents?
3∙3∙3∙3∙3
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 35
Example 2 -You try!
• Rewrite each expression with exponents
1.
(5 + 𝑎)(5 + 𝑎) (5 + 𝑎) (5 + 𝑎)
2.
8∙8∙8∙3∙3∙3∙3
Answers
1. (5 + 𝑎)(5 + 𝑎) 5 + 𝑎 5 + 𝑎 = 𝟓 + 𝒂
2.
8 ∙ 8 ∙ 8 ∙ 3 ∙ 3 ∙ 3 ∙ 3 = 𝟖𝟑 ∙ 𝟑𝟒
𝟒
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…
x
• y = 120,000 1.06
• How do we find the value after 3 years?
• We know x represents years, so plug in
3 for x.
• y = 120,000 1.06 3
• 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/decay
r = 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 = 10,000 1.03
• y= 11592 bugs
5
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 = 18000 .93
• y= 12,522.39
5
Homework
6.2 Worksheet
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Problems:
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