1. 9.1 Exponential Functions

2. Exponential Functions
A function of the form y=abx,
where a=0, b>0 and b=1.
Characteristics
1. continuous and one-to-one
2. domain is the set of all real numbers
3. Range is either all real positive numbers or all real negative numbers depending on whether a is < or > 0
4. x-axis is a horizontal asymptote
5.y-intercept is at a
6. y=abx and y=a(1/b)x are reflections across the y-axis

3. Example 1
Sketch the graph of y=2x. State the domain and range.

4. Example 2
Sketch y=( )x. State the domain and range.

5. Exponential Growth & Decay
Exponential function with base greater than one.
y=2(3x)
Exponential Decay:
Exponential function with base between 0 and 1
y=4(1/3)x

6. Example 3-6 Determine if each function is exponential growth or decay
y=(1/5)x y=7(1.2)x
y=2(5)x y=10(4/3)x

7. Steps to write an exponential function
1. Use the y-intercept to find a
2. Choose a second point on the graph to substitute into the equation for x and y. Solve for b.
3. Write your equation in terms of y=abx (plug in a and b)

8. Example 7
Write an exponential function using the points (0, 3) and (-1, 6)

9. Example 8
Write an exponential function using the points (0, -18) and (-2, -2)

10. Example 9
In 2000, the population of Phoenix was 1,321,045 and it increased to 1,331,391 in 2004.
A. Write an exponential function of the form y=abx that could be used to model the population y of Phoenix. Write the function in terms of x, the number of years since 2000.
B. Suppose the population of Phoenix continues to increase at the same rate. Estimate the population in 2015.

11. Exponential Equations
An equation in which the variables are exponents
Property of Equality
If the base is a number other than 1 and the base is the same , then the two exponents equal each other.
2x = 28 then x=8

12. Steps to Solve Exponential Equations/inequalities
1. Rewrite the equation so all terms have like bases (you may need to use negative exponents)
2. Set the exponents equal to each other
3. Solve
4. Plug x back in to the original equation to make sure the answer works

13. Example 10
Solve n+1 = 81

14. Example 11
Solve x = 92x-1

15. Example 12
Solve x = 8x-1

16. Example 13
Solve

17. Example 14
Solve

18. Example 15
Solve