1. Exponential Functions and their Graphs
Section 3.1

2. Exponential Functions & Graphs
What is an exponential function?
A function with base a denoted by:
where a > 0, a ≠ 1, and x is any real
number

3. Exponential Functions & Graphs
Notice that with exponential functions, the variable is the exponent.
In algebraic functions (i.e. x²) the variable is the base

4. Exponential Functions & Graphs
Compute the following exponential for the following values of x.
f (x) =
f (x) =
f (x) = 216
f (x) = 0.005

5. Exponential Functions & Graphs
Fill in the following table for the two exponential functions. X -2 -1 1 2 3
Using these points, graph the curves for both exponential functions on the same coordinate axis.

6. Exponential Functions & Graphs

7. Exponential Functions & Graphs
Now on the same coordinate axis, graph the following two functions. X -2 -1 1 2

8. Exponential Functions & Graphs

9. Exponential Functions & Graphs
From these graphs, we notice that for:
→ horizontal shift of -z
→ vertical shift of z
→ reflection over the x-axis
→ reflection over the y-axis

10. Exponential Functions & Graphs
The Natural Base e
Irrational number, similar to pi
Approximately equal to …..
When e is the base of an exponential function, the function is called the natural exponential function

11. Exponential Functions & Graphs
Evaluate and graph the function at the indicated values. X -2 -1 1 2
0.135
0.368 1
2.718
7.389

12. Exponential Functions & Graphs

13. Exponential Functions & Graphs
The number e is an important number that comes up frequently in finance
It is used to evaluate continuously compounding interest
Compound interest is found by using the formula:

14. Exponential Functions & Graphs
As x →∞,
f(x) → e

15. Exponential Functions & Graphs
One of the most common exponential equations is the formula for compound interest.
Compound interest means that you are earning interest on your original principal and any interest earned.
Using the formula saves you from doing several calculations for 1 particular problem.

16. Exponential Functions & Graphs
In general, the account balance of an investment compounded “n” times per year for “t” years is given by:
A = account balance
r = interest rate
P = initial Principal
t = number of years
n = number of compounds
per year

17. Exponential Functions & Graphs
Suppose you invest $1,000 into an account that has an annual interest rate of 3% compounded monthly. How much would be in the account after 5 years?
P =
1,000
t = 5
r =
0.03
n = 12

18. Exponential Functions & Graphs
Suppose you invest $1,000 into an account that has an annual interest rate of 3% compounded monthly. How much would be in the account after 5 years?
A =
1,161.62

19. Exponential Functions & Graphs
An investment of $5,000 is made into an account that pays 6% annually for 10 years. Find the amount of money in the account if interest is compounded:
Annually:
Semiannually:
Monthly:
Daily:
$8,854.24
$9,030.56
$9,096.98
$9,110.14

20. Exponential Functions & Graphs
Notice that as the number of compounds per year increases, the account balance increases.
If we let the number of compounds increase without boundaries, the process approaches what is called continuous compounding.
This means that your interest is always being compounded.

21. Exponential Functions & Graphs
In the formula what happens as x approaches infinity.
Therefore, the formula for continuously compounding interest is:

22. Exponential Functions & Graphs
Find the amount in an account after 10 years if $6,000 is invested at an interest rate of 7% compounded continuously.
r =
0.07
P =
6,000
t = 10

23. Exponential Functions & Graphs
Determine the balance at the end of 20 years if $1,500 is invested at 6.5% interest and the interest is compounded:
Quarterly:
Daily:
Continuously:
$5,446.73
$5,503.31
$5,503.95

24. Exponential Functions & Graphs
A total of $12,000 is invested at an annual rate of 9%. Find the balance after 5 years if the interest is compounded:
Quarterly:
Monthly:
Continuously:
$18,726.11
$18,788.17
$18,819.75

25. Exponential Functions & Graphs
Determine the amount of money that should be invested at 9% interest, compounded monthly, to produce a final balance of $30,000 after 15 years.
P =
???
t =
A = 15
$30,000
r =
0.09
n = 12

26. Exponential Functions & Graphs

27. Exponential Functions & Graphs

28. Exponential Functions & Graphs

29. Exponential Functions & Graphs

30. Exponential Functions & Graphs

31. Exponential Functions & Graphs

32. Exponential Functions & Graphs

33. Exponential Functions & Graphs

34. Exponential Functions & Graphs