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Published byDominick Stewart Modified over 9 years ago
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Warm up A rabbit population starts with 3 rabbits and doubles every month. Write a recursive formula that models this situation. What is the number of rabbits after 6 months?
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Exponential Functions
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Recursions to Exponentials
We can write geometric recursive formulas as exponential functions. This is nice because it allows us to find the next terms without knowing the previous.
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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 of the exponential function is π¦=π π π₯
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Look at warm up to determine what the variables mean
π¦=π π π₯ Letβs determine how many rabbits there are in the first 3 months. Month 0 is the starting amount. π¦=3 (2) π₯ As we can see: a= starting number b= rate of change x= number of time intervals that have passed. Month Number of rabbits 3 1 3 β 2 =6 2 3 β 2β2=12 3 β 2 β 2β2=24
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Example 1 A house was purchased for $120,000 and is expected to increase in value at a rate of 6% per year. First write a recursive sequence modeling the value of the house. Then write an exponential function modeling the value.
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Example 1: Solution A house was purchased for $120,000 and is expected to increase in value at a rate of 6% per year. Recursion: π 0 =120,000 π π =1.06 π πβ1 Exponential: π¦=120, π₯
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Looking at the βbβ in another way:
Decay: if b is less than 1 Growth: If b is greater than 1 aΒ = initialΒ amountΒ before measuring growth/decay rΒ = growth/decayΒ rateΒ (often a percent) xΒ = number ofΒ timeΒ intervals that have passed
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Example 2 Is the following equation modeling growth or decay?
π π₯ = π₯ Decay π π₯ = π₯ Growth
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Example 3 Sometimes we will need to determine how much growth or decay there is given values. Find the percent increase or decrease in the numbers below: 36,32
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Example 3 - solution 36,32 First find the ratio between the two by doing π πππππ ππππ π‘ = = .88 Since the second number is smaller than the first, we know it is decay. To find percent of decay do = .11 So the percent of decrease is 11%.
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Example 4 Do we do the same method if the numbers are increasing?
Try and see: 63, 100.8
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Example 4 Solution 63, 100.8 π πππππ ππππ π‘ = 100.8 63 =1.6
The 1 represents the whole number, and the 6 represents the percent change, So these numbers have a 60% increase.
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This is a big assignment! Do not put it off!
Homework 5.1 Worksheet Problems: 1 a-c 2 4 a-d 5 a-c 6 This is a big assignment! Do not put it off!
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