1. Chapter 7 – Exponential and logarithmic functions
7.1 – Exploring Exponential Models

2. 7.1 – Exploring Exponential Models
#Goals:
To model exponential growth and decay

3. 7.1 – Exploring Exponential Models
Definition
A function with the general form y = abx
Facts/Characteristics
a ≠ 0
b > 0
b ≠ 1
Examples
Non-Examples
Exponential Function

4. 7.1 – Exploring Exponential Models
Example 1
What is the graph of y = 3.1x

5. 7.1 – Exploring Exponential Models
Got it 1?
What is the graph of y = 2(3)x

6. 7.1 – Exploring Exponential Models
Definition
As the value of x increases, the value of y increases
Facts/Characteristics
Examples
Non-Examples
Exponential Growth

7. 7.1 – Exploring Exponential Models
Definition
As the value of x increases, the value of y decreases, approaching 0
Facts/Characteristics
Examples
Non-Examples
Exponential Decay

8. 7.1 – Exploring Exponential Models
Definition
A line that a graph approaches as x or y increases in absolute value
Facts/Characteristics
Examples
Non-Examples
Asymptote

9. 7.1 – Exploring Exponential Models
Exponential Function
For the function y = abx
a > 0
b > 1
Exponential Growth
0 < b < 1
Exponential Decay
Domain: all reals y-intercept: (0, a)
Range: {y|y > 0} Asymptote: y = 0

10. 7.1 – Exploring Exponential Models

11. 7.1 – Exploring Exponential Models
Example 2
Identify y = 0.7x as an example of exponential growth or decay. What is the y-intercept?

12. 7.1 – Exploring Exponential Models
. Got it:
What is the graph of y = 4( )x

13. 7.1 – Exploring Exponential Models
Definition
The value of b when b > 1
Facts/Characteristics
Examples
Non-Examples
Growth Factor

14. 7.1 – Exploring Exponential Models
Definition
The value of b when 0 < b < 1
Facts/Characteristics
Examples
Non-Examples
Decay Factor

15. 7.1 – Exploring Exponential Models
Growth Factor
Increases by constant percentage each time period
percent increase r
rate of increase
growth rate
b = 1 + r
Decay Factor
Decreases by constant percentage each time period
percentage decrease
rate of decay

16. 7.1 – Exploring Exponential Models

17. 7.1 – Exploring Exponential Models
Example 3
You buy a savings bond for $25 that pays a yearly interest rate of 4.2%. What will the savings bond be worth after fifteen years?

18. 7.1 – Exploring Exponential Models
Got it 3?
Suppose you deposit $2000 in a savings account that pays interest at an annual rate of 4%. If no money is added or withdrawn from the account, how much will be in the account after 18 years?

19. 7.1 – Exploring Exponential Models
Example 4
You open a savings account that pays 4.5% annual interest. If your initial investment is $300 and you make no additional deposits or withdrawals, how many years will it take for the account to grow to at least $500?

20. 7.1 – Exploring Exponential Models
Got it 4?
Suppose you deposit $2000 in a savings account that pays interest at an annual rate of 4%. If no money is added or withdrawn from the account, how many years will it take for the account to contain $3000?

21. 7.1 – Exploring Exponential Models
Exponential functions are often discrete (individual points)
There is never exactly $3000 in the account
To model a discrete situation using an exponential function in the form y = abx, you need to find the growth or decay factor b
If you know y-values for 2 consecutive x values, you can find the rate of change (r) and then find b
and b = 1 + r

22. 7.1 – Exploring Exponential Models
Example 5
The initial value of a car is $30,000. After one year, the value of the car is $20,000. Estimate the value of the car after five years.

23. 7.1 – Exploring Exponential Models
Got it 5?
A population of 1,860,000 decreases 1.5% each year for 12 years. Write an exponential function to model the situation. Find the amount after 12 years.

24. 7.1 – Exploring Exponential Models
HOMEWORK Page 439 #10 – 28 (even)