1. Properties of Exponentials Functions.
What you’ll learn
To explore the properties of functions of the
form
To graph exponential functions that have
base e.
Vocabulary
Natural base exponential function,
Continuously compounded interest

2. You can apply the four types of transformations
Take a note:
You can apply the four types of transformations
-stretches, compressions, reflections, and translations-
to exponential functions. The factor a in
can stretch or compress and possible reflects the
graph of the parent function
For example: x -2 -1 1 2

3. How does the graph of compare to the graph of the parent function? x
Problem 1: Graphing
How does the graph of compare to
the graph of the parent function? x -2 -1 1 2

4. Your turn
How does the graph of compare
to the graph of the parent function?
Answer:
Reflects across the x-axis;
compress by a factor of 0.5

5. Problem 2 Translating the Parent Function
How does the graph of each function compare to
the graph of the parent function?
Graph then translate
4 units to the right
Make the table of values x -1
½=0.5 1 2 4 3 8

6. x Graph then translate 10 units up. (x,y) Make a table of values -1 140
(-1,40)
(-1,50) 20
(0,20)
(0,30) 1 10
(1,10)
(1,20) 2 5
(2,5)
(2,15) 3
2.5
(3,2.5)
(3,12.5)

7. Your turn
How does the graph of each function compare to the
graph of the parent function?
Translate 2 units to the left;
the y-intercepts becomes 16
Answers:
b) Stretch the graph of by a factor of 5
And translate the graph of units.

8. Problem 3: Using an exponential Model
The best temperature to brew coffee is between 1950F and 2050F. Coffee is cool enough to drink at 1850F. The table shows temperature readings from a sample cup of coffee. How long does it take for a cup of coffee to be cool enough to drink? Use an exponential model. The room temperature is 68 degrees
Step 1: plot the date to determine if an exponential model is realistic.
STAT then 1 then ENTER plot the data in L1 and L2.
Time
Temp
203 5
177 10
153 15
137 20
121 25
111 30
104
Step 2: the graphing calculator model assumes
the asymptote is y=0. Since room temperature
is about 680F, Subtract 68 from each temperature
value. Calculate the third list by letting L3=L2-68
Step 3: Use the ExpReg L1,L3 (from STAT then to the
right to CALC then 0 then ENTER) function on the
transformed data to find en exponential model.
Step 4: Translate vertically by 68 units to model the original data.
Use the model to find how long it takes to the coffee to cool
to 1850F.Use the y function and plug it in.

9. Your turn
Use the exponential model. How
long does it take for the coffee to
reach a temperature of 100 degree F.
Answer:
About 31.9 min(go to the table and look for it)
b) In Problem 3, would the model of the exponential
Data be useful if you did not translate the data by
68 units? Explain
Answer
No; a hot coffee cannot cool below room temperature.
So, to use exponential data, it is important to
translate the data by 68 units

10. Take a note
Read your note about “e”

11. Amount in account at time t
Problem 4: Evaluating
How can you use a graphing calculator to evaluate ?
In the lesson before we learned the interest that was
Compound annually. The formula for continuously
compounded interest uses the number e.
Interest rate annual
Amount in account at time t
Time in years
Principal

12. Problem 4: Continuously Compounded Interest
Suppose you won a contest at the start of 5th
grade that deposited $3000 in an account that pays
5% annual interest compounded continuously. How much will you have in the account when you enter high school 4 years later?
Your turn: About how much will be in the account
After 4 years of high school?
Answer
$4475

13. Classwork odd Homework even
TB pgs exercises 7-42