Aim: Exponential Function Course: Math Literacy Aim: How does the exponential model fit into our lives? Do Now: The price of an item increased by 18% due to high demand. The amount of increase was $95. What was the original price of the item?

Aim: Exponential Function Course: Math Literacy Zero Power Property Properties of Exponents Product of Powers Property a 0 = 1 Power of Power Property Power of Product Property Negative Power Property Quotients of Powers Property Power of Quotient Property a m a n = a m+n (a m ) n = a mn a -n = 1/a n, a  0 (ab) m = a m b m

Aim: Exponential Function Course: Math Literacy Types of exponents Positive Integer Exponent a n = a a a a n factors Zero Exponent a 0 = 1 Rational Exponent 1/n Rational Exponent m/n Negative Exponent - m/n beware! -2 3/2 is not the same as (-2) 3/2

Aim: Exponential Function Course: Math Literacy Exponential Function y = a b x.. have variables as exponents, and where a  0, base b > 0, and also b  1. The x-axis is a horizontal asymptote: a x  0 as x  -  The domain (x) is the set of real numbers: (- ,  ) The range (y) is the set of positive real numbers: (0,  ) If b > 1, the graph is increasing and continuous As b increases in value from 1, the slope of the graph gets steeper y = 1 2 x y = 2 x When a = 1, graph always goes through (0,1) (0,1) What happens as b increases in value?

Aim: Exponential Function Course: Math Literacy The b Affect y = 2 x y = (1/2) x (0,1) the graph is decreasing - Decay If b is a positive number other than 1, the graphs of y = b x and y = (1/b) x are reflections through the y-axis of each other If b > 1, the graph is increasing - Growth What if b = 1? horizontal line: y = 1 y = 1 y = a b x If 0 < b < 1

Aim: Exponential Function Course: Math Literacy The b Affect: b < 0 y = a b x Let b = (-2)? What if b < 0? if b < 0, no longer the exponential function y = 1 (-2) x graph: table:  x = 1table:  x =.1

Aim: Exponential Function Course: Math Literacy The a Affect Graph f(x) = -(1/2) x = -1 (1/2) x Graph f(x) = -6 x = -1 (6) x (0,-1) y = a b x Graph f(x) = -2 x = -1 2 x

Aim: Exponential Function Course: Math Literacy The a Affect (0,4) a = 4 (0,2) a = 2 y = a b x (0,1) a = 1

Aim: Exponential Function Course: Math Literacy The a Affect Graph the exponential functions

Aim: Exponential Function Course: Math Literacy Model Problem The exponential function f(x) = 13.49(0.967) x – 1 describes the number of 0-rings expected to fail, f(x), when the temperature is x o F. On the morning the Challenger was launched, the temperature was 31 o F, colder than any previous experience. Find the number of 0-rings expected to fail at this temperature. x o = 31

Aim: Exponential Function Course: Math Literacy Model Problem Horses were born with eight deformed legs, pigs with no eyes, and eggs contained several yolks. This was part of the grotesque aftermath of the 1986 explosion at the Chernobyl nuclear power plant in the former Soviet Union. Nearby cities were abandoned and 335,000 people were permanently displaced from their homes. The explosion sent about 1000 kilograms of radioactive cesium-137 into the atmosphere. The function f(x) = 1000(0.5) x/30 describes the amount of cesium-137, f(x), in kilograms, remaining in Chernobyl x years after If even 100 kilograms of cesium-137 remain in Chernobyl’s atmosphere, the area is considered unsafe for human habitation. Will people be able to live in the area 80 years after the accident?

Aim: Exponential Function Course: Math Literacy Model Problem Horses were born with eight deformed legs, pigs with no eyes, and eggs contained several yolks. This was part of the grotesque aftermath of the 1986 explosion at the Chernobyl nuclear power plant in the former Soviet Union. Nearby cities were abandoned and 335,000 people were permanently displaced from their homes. The explosion sent about 1000 kilograms of radioactive cesium-137 into the atmosphere. The function f(x) = 1000(0.5) x/30 describes the amount of cesium-137, f(x), in kilograms, remaining in Chernobyl x years after If even 100 kilograms of cesium-137 remain in Chernobyl’s atmosphere, the area is considered unsafe for human habitation. Will people be able to live in the area 80 years after the accident? f(x) = 1000(0.5) x/30 f(x) = amount of cesium remaining; x = 80 yrs f(80) = 1000(0.5) 80/30 f(80)  157 kilograms habitation not possible