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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.

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Presentation on theme: "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."β€” Presentation transcript:

1 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?

2 Exponential Functions

3 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.

4 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 𝑦=π‘Ž 𝑏 π‘₯

5 Look at warm up to determine what the variables mean
𝑦=π‘Ž 𝑏 π‘₯ Let’s determine how many rabbits there are in the first 3 months. Month 0 is the starting amount. 𝑦=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 βˆ™ 2 =6 2 3 βˆ™ 2βˆ™2=12 3 βˆ™ 2 βˆ™ 2βˆ™2=24

6 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.

7 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‘ˆ 0 =120,000 π‘ˆ 𝑛 =1.06 π‘ˆ π‘›βˆ’1 Exponential: 𝑦=120, π‘₯

8 Looking at the β€œb” 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

9 Example 2 Is the following equation modeling growth or decay?
𝑓 π‘₯ = π‘₯ Decay 𝑓 π‘₯ = π‘₯ Growth

10 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

11 Example 3 - solution 36,32 First find the ratio between the two by doing π‘ π‘’π‘π‘œπ‘›π‘‘ π‘“π‘–π‘Ÿπ‘ 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%.

12 Example 4 Do we do the same method if the numbers are increasing?
Try and see: 63, 100.8

13 Example 4 Solution 63, 100.8 π‘ π‘’π‘π‘œπ‘›π‘‘ π‘“π‘–π‘Ÿπ‘ 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.

14 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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