1. 8.7 Exponential Functions

2. Goals / “I can….” Evaluate exponential functions Graph exponential functions 8.7 – Exponential Functions

3. Definition: An exponential function is a function in the form y = a * b where a is the starting value b is the growth factor x is the power. x

4. First, let’s take a look at an exponential function xy 01 12 24 1/2 -21/4 8.7 – Exponential Functions

5. So our general form is simple enough. The general shape of our graph will be determined by the exponential variable. Which leads us to ask what role does the ‘a’ and the base ‘b’ play here. Let’s take a look. 8.7 – Exponential Functions

6. b First let’s change the base b to positive values What conclusion can we draw ? 8.7 – Exponential Functions

7. b 0<b<1 Next, observe what happens when b assumes a value such that 0<b<1. Can you explain why this happens ? 8.7 – Exponential Functions

8. What do you think will happen if ‘b’ is negative ? 8.7 – Exponential Functions

9. Don’t forget our definition ! Can you explain why ‘b’ is restricted from assuming negative values ? Any equation of the form: 8.7 – Exponential Functions

10. a To see what impact ‘a’ has on our graph b3 we will fix the value of ‘b’ at 3. What does a larger value of ‘a’ accomplish ?

11. Shall we speculate as to what happens when ‘a’ assumes negative values ? Let’s see if you are correct ! 8.7 – Exponential Functions

13. Our general exponential form is “b” is the base of the function and changes here will result in: When b>1, a steep increase in the value of ‘y’ as ‘x’ increases. When 0<b<1, a steep decrease in the value of ‘y’ as ‘x’ increases. 8.7 – Exponential Functions

14. To evaluate an exponential function…… domain value Put the domain value (the given #s) in for the exponent.

15. 8.7 – Exponential Functions The domain is {-2, 0, 3}. Evaluate the function. y = 4xy = 10 * 5 x

16. 8.7 – Exponential Functions Suppose 20 rabbits are taken to an island. The rabbit population then triples every half year. The function f(x) = 20 * 3, where x is the number of half – year periods. How many rabbits would there be after 2 years? x

17. 8.7 – Exponential Functions Graphs of Exponential Functions: Use your calculator. y = 0.5 * 2y = - 0.5 * 2 xx

18. 8.7 – Exponential Functions a What is the difference in the graph if the “a” value is positive or negative?