1. Exploring Exponential Models.
What you’ll learn
To define exponential functions.
Graph exponential functions.
Solve applications of exponential growth and decay.
Vocabulary
Exponential function, exponential growth, exponential decay, asymptote, growth factor, decay factor

2. Take a note:
An exponential function is a function with the general
form , In an exponential function the base b is a constant. The exponent x is the independent variable with domain the set of real numbers.
Examples:

3. Problem 1:Graphing an Exponential Function.
What is the graph
Step 1: Make a table of values.
Step 2: Plot and connect the points. 𝑥 𝑦 -4 -3 -2 -1 1 2 3

4. What is the graph of each function?
Your turn:
What is the graph of each function?
Exponential
Decay
Exponential
Growth
The 𝑥−𝑎𝑥𝑖𝑠 is
an asymptote -2 9 -1 3 1
0.3 -2
0.06 -1
0.25 1 4

5. Notes from the vocabulary
There are two types of exponential behavior:
exponential growth and exponential decay.
For exponential growth, as the value of x increases,
the value of y increases.
For exponential decay, as the value of x increases,
the value of y decreases, approaching zero.
The exponential functions shown here are
asymptotic to the axis. An asymptote is a line that a
graph approaches as x increases or y increases in
absolute value.
Go back to the previous slide

6. Problem 2: Identifying Exponential Growth and Decay
Take a note
Problem 2: Identifying Exponential Growth and Decay
Identify each function or situation as an example
of exponential growth or decay. What is the y-intercept?
a) 0<b<1 decay (0,12)
Answers
b) b>1 growth (0,0.25)
c) 5% is the growth, y-intercept is 1000 is the initial investment

7. Your turn
Identify each function or situation as an example of exponential growth or decay. What is the y –intercept?
Answers
Exponential growth;3
b) Exponential decay; 11
c) Exponential growth; 2000

8. Amount after t time periods
A very important note:
For the function
If b>1 then is an
exponential growth and
b is the growth factor.
A quantity that exhibits
exponential growth
increases by a constant
percentage each period of time.
That percentage increase r,
written as a decimal is the
rate of increase or growth
rate.b=1+r
If 0<b<1 then is an
Exponential decay and
b is the decay factor.
The quantity decreases
by a constant percentage
each period of time. The
percentage decrease r,
is the rate of decay.
Usually a rate of decay is
expressed as a negative
quantity, so b=r+1
Rate of growth
Initial amount
Amount after t time periods
Number of time periods

9. Answer Problem 3:Modeling Exponential Growth.
You invested $1000 in a savings account at the end of 6th grade.
The account pays 5% annual interest. How much money will be in the account after six years?
Answer
Step 1: Determine if an exponential function is a reasonable model.
The money grows at a fixed rate of 5% per year.
An exponential model is appropriate.
Step 2: Define the variables and determine the model.
t= the number of years since the money was invested
A(t)= the amount in the account each year.
Step 3: Use the model to solve the problem

10. Your turn
Suppose you invest $500 in a savings account that pays 3.5% annual interest. How much will be in the account after 5 years?
Answer:

11. Answer Problem 4: Using Exponential Growth:
Suppose you invest $1000 in a savings account that pays 5% annual interest.
If you make no additional deposits or withdrawals, how many years will it takes for the account to growth to at least $1500?
Answer
Step 1: Define the variables
t= number of years and
A(t)=the amount in the account after t years
Step 2: Determine the model
Step 3: Make a table
Get the graphing calculator, made the table using the table feature. Find the input when the output is 1500.
Answer: The acc. will no contain $1500 until the 9th year.

12. Your turn
a)Suppose you invest in a savings account that pays 3.5% annual interest. When will the account contains at least $650.
b) Use the table in Problem 4 to determine when that account will
contain at least $1650.
Answers
b) After 11 years; the account contains $

13. Take a note: Exponential forms are often discrete
(meaning discrete graph is just points, and the points
are not connected in any way). Like for example
problem 4, interest is paid only once a year so the
graphs consist of individual points when t=1, t=2, t=3
and so on. To model a discrete situation using the
exponential form you will need to find
the growth or decay for factor b. So you will need to
find the rate of change(remember you know the
y-value for two consecutives x-values)

14. Problem 5: Writing an Exponential Form.
The table shows the world population of Iberian Lynx in 2003 and 2004 Is this trend continues and the population is decreasing exponentially, how many Iberian lynx will be there in 2014?
World population
Of Iberian Lynx
Year
2003
2004
Population
150
120

15. Step 1: Using the general form . Define the variables.
x= the numbers of years since 2003
Y= the population of the Iberian Lynx.
Step 2: Determine r
Step 3: Use r to find b:
Step 4: Write the model
Step 5: Use the model to
find the population in2014

16. Your turn:
a)For the model in Problem 5, what will be The world population of Iberian lynx in 2020?
b) If you graphed the model in Problem 5, will it ever cross the x-axis? Explain
Answer: a)
b) no, the function is asymptotic to the x-axis

17. CW/HW
Form G