13.1/13.2 - Exponential Growth and Decay Functions

Definition An exponential function is a function with the general form 𝑦=𝑎 𝑏 𝑥 , 𝑎≠0, and 𝑏≠1. Base = b

Ex: Graph 𝑦= 2 𝑥 . Identify a and b. -3 -2 -1 1 2 3 Domain: _________________ Range: __________________ What do you notice about the y-intercept? What value is the function increasing by each time?

Ex: Graph 𝑦= 1 2 𝑥 . Identify a and b. Domain: _________________ Range: __________________ x y -3 -2 -1 1 2 3 What do you notice about the y-intercept? What value is the function increasing by each time?

Exponential Growth/Decay

Note: Last class we used the formula 𝑦=𝑎 (1+𝑟) 𝑡 for exponential growth and 𝑦=𝑎 (1−𝑟) 𝑡 for exponential decay. It should make sense now why, when we added the rate, r, to 1, we had exponential growth. When we subtracted the rate, r, from 1, we had exponential decay. So anytime b is more than 1, it’s because we had exponential growth. Anytime b is less than 1, it’s because we had exponential decay.

Exponential Growth and Decay https://www.youtube.com/watch?v=GGytywqpGXA Stop at 7:44

Ex: Identify each function or situation as an example of exponential growth or decay. What is the y-intercept? 𝑦=12 (0.95) 𝑥 𝑦=0.25 (3) 𝑥 You invested $1000 in a college savings account at the end of 6th grade. The account pays 5% interest annually.

Growth Factor/Decay Factor Always b Growth factor if exponential growth Decay factor if exponential decay Note: Last class we used the formula 𝑦=𝑎 (1+𝑟) 𝑡 for exponential growth and 𝑦=𝑎 (1−𝑟) 𝑡 for exponential decay. But 1±𝑟 is the same as 𝑏.

Ex:

Transformations 𝑦=𝑎 𝑏 (𝑥−ℎ) +𝑘 Parent function: 𝒚= 𝒃 𝒙 𝑎<0: reflection over x-axis 𝑎>1: Stretch 0<𝑎<1: Compress Vertical Translation by k Horizontal Translation by h

Ex: How does the graph of 𝑦=− 1 3 ∙ 3 𝑥 compare to the graph of the parent function? x y -3 -2 -1 1 2 3 HA: __________ Y-int: _________ Domain: _________________ Range: __________________

Why do you think I started with 4 in the middle? Ex: How does the graph of 𝑦= 2 (𝑥−4) compare to the graph of the parent function? x y 1 2 3 4 5 6 7 HA: __________ Y-int: _________ Domain: _________________ Range: __________________ Why do you think I started with 4 in the middle?

Ex: How does the graph of 𝑦=20 1 2 𝑥 +10 compare to the graph of the parent function? x y -3 -2 -1 1 2 3 HA: __________ Y-int: _________ Domain: _________________ Range: __________________

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