1. 8.2 – Properties of Exponential Functions

2. Review: what is an asymptote?
“Walking halfway to the wall”
An Asymptote is a line that a graph approaches as x or y increases in absolute value.
In this example, the asymptote is the x axis.

3. Graphing y=abx when a<0
Ex: Graph
Sketch your prediction of what the graph will look like
Where is the asymptote?

4. Translating y=abx
How does the equation change if we want to move both graphs up 4 units? Predictions?
Question: where is the asymptote now?
To move the graph up or down, add or subtract units at the end of the equations. No need to use inverses – if you want to go up, add; if you want to go down, subtract.

5. Translating y=abx
How does the equation change if we want to move both graphs left 4 units? Predictions?
To move the graph left or right, add or subtract units to the exponent of the equation. Reminder: use the inverse of how you want the graph to move (e.g. x-4 will move to the right; x+4 will move to the left)

6. Let’s try some Graph each function as a translation of y=9(3)x
Make a table of values for each
Graph, from -3 to 3

7. y =9(3)x+1
y =9(3)x-4-1
y =9(3)x-4

8. e is an irrational number, approximately equal to 2.718.
What is base “e” ?
e is an irrational number, approximately equal to
Exponential functions with a base of e are useful for describing continuous growth or decay. In the graph below, y = e is the asymptote to the graph.
y = e

9. Graphing ex
Using your graphing calculators, graph y=ex. Evaluate e4 to four decimal places.
We now need to evaluate where x=4

10. 2. Press 2nd, Calc and select 1 (value). Press enter
3. We are evaluating when x=4. Enter 4 for x and press enter.

11. The value of e4 is about 54.59815 Your turn: evaluate e-3
0.0498
So, what is “e” good for???

12. Continuously Compounding Interest
A = Pert
A = amount of money in the account
P = principal (how much is deposited)
r = annual rate of interest
t = time (in years)

13. Example: Continuously Compounded Interest Problem
You invest $1,050 at an annual interest rate of 5.5%, compounded continuously. How much will you have in the account after 5 years?
Start with:
A = Pert
1050(e)0.055(5)
1050(e)0.275
1050( )
A = $
P=$1050, r=5.5% = 0.055, t=5
Substitute in for p, r, and t
Simplify they power
Evaluate e0.275 with your calculator
Simplify

14. Let’s try one:
Suppose you invest $1,300 at an annual interest rate of 4.3%, compounded continuously. How much will you have in the account after three years?

15. Suppose you invest $1,300 at an annual interest rate of 4
Suppose you invest $1,300 at an annual interest rate of 4.3%, compounded continuously. How much will you have in the account after three years?