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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 = 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. Month Number of rabbits 0 3 1 3∙2=6 2 3 ∙ 2∙ 2 = 12 3 3 ∙ 2 ∙ 2∙ 2 = 24 = 3(2) 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 + )(5 + ) (5 + ) (5 + ) 2. 8∙8∙8∙3∙3∙3∙3 Answers 1. (5 + )(5 + ) 5 + 5 + = + 2. 8 ∙ 8 ∙ 8 ∙ 3 ∙ 3 ∙ 3 ∙ 3 = ∙ 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. = 120,000 1.06 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 • = 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 • = 18000 .93 • y= 12,522.39 5 Homework 6.2 Worksheet • • • • • Problems: 1 2 3 6