Download Lecture notes for Section 9.2 (Exponential Functions)

Int. Alg. Notes
Section 9.2
Page 1 of 6
Section 9.2: Exponential Functions
Big Idea: The exponential function is a base raised to a power that is a variable.
Big Skill: You should be able to graph an exponential function, solve basic exponential equations, and use
exponential function models.
Definition: Exponential Function
An exponential function is a function of the form
f  x  ax
where a is a positive real number (i.e., a > 0) and a  1. The domain of the exponential function is the set of all
real numbers.
The exponential function is unlike anything we’ve seen before in algebra because hypothetically we only know
how to work with rational exponents, and the exponential function has an exponent that can be any real number,
including an irrational number. However, it turns out that irrational exponents are well-defined (in calculus).
Practice:
1. Use a calculator to evaluate 51.4 
2. Use a calculator to evaluate 51.41 
3. Use a calculator to evaluate 51.414 
4. Use a calculator to evaluate 51.4142 
5. Use a calculator to evaluate 5
2

Algebra is:
the study of how to perform multi-step arithmetic calculations more efficiently,
and the study of how to find the correct number to put into a multi-step calculation to get a desired answer.
Int. Alg. Notes
Section 9.2
Page 2 of 6
To graph exponential functions, we will compute a table of points, then connect the points with a smooth curve.
You will notice that all exponential function graphs look kind of the same when the base is greater than 1.
Practice:
6. Graph the exponential function f  x   2x
x
y  f  x   2x
(x, y)
-3
-2
-1
0
1
2
3
7. Graph the exponential function f  x   3x
x
y  f  x   3x
(x, y)
-3
-2
-1
0
1
2
8. Graph the exponential function f  x   10x
x
y  f  x   10x
(x, y)
-3
-2
-1
0
1
Algebra is:
the study of how to perform multi-step arithmetic calculations more efficiently,
and the study of how to find the correct number to put into a multi-step calculation to get a desired answer.
Int. Alg. Notes
Section 9.2
Page 3 of 6
Properties of the Graph of an Exponential Function, f  x   a x , when a > 1





The domain is the set of all real numbers.
The range is the set of all positive real numbers.
There are no x-intercepts.
The y-intercept is 1.
1

The graph of f contains the points  1,  ,  0,1 , and 1, a  .
a

Now look at the graphs of exponential functions when the base is less than 1:
We will next see that the following properties hold true for the graph of an exponential function when the base
is less than 1:
Properties of the Graph of an Exponential Function, f  x   a x , when a < 1






The domain is the set of all real numbers.
The range is the set of all positive real numbers.
There are no x-intercepts.
The y-intercept is 1.
1

The graph of f contains the points  1,  ,  0,1 , and 1, a  .
a

The graph is a reflection about the y-axis of the graph one obtains when a > 1.
x

  1 1   1   x
The function can be written as f  x   a        
 a    a 


x
Algebra is:
the study of how to perform multi-step arithmetic calculations more efficiently,
and the study of how to find the correct number to put into a multi-step calculation to get a desired answer.
Int. Alg. Notes
Section 9.2
Page 4 of 6
Practice:
1
9. Graph the exponential function f  x    
2
x
1
y  f  x   
 2
x
x
(x, y)
-3
-2
-1
0
1
2
3
1
10. Graph the exponential function f  x    
 3
x
1
y  f  x   
 3
x
x
(x, y)
-2
-1
0
1
2
3
 1
11. Graph the exponential function f  x    
 10 
x
1
y  f  x   
 10 
x
x
(x, y)
-1
0
1
2
3
Algebra is:
the study of how to perform multi-step arithmetic calculations more efficiently,
and the study of how to find the correct number to put into a multi-step calculation to get a desired answer.
Int. Alg. Notes
Section 9.2
Page 5 of 6
Definition
The number e (Euler’s Number) is defined as the number that the expression
n
 1
1  
 n
approaches as n gets bigger and bigger. Approximately, e  2.718 281 827
Property for Solving Exponential Equations
If a u  a v , then u = v.
Practice:
12. Solve 5 x 4  51
13. Solve 2 x1  16
2
3x
14. Solve x  27
9
Applications of Exponential Functions: Exponential Probability
Practice:
15. The manager of a crisis helpline knows that from 3:00 AM to 5:00 AM, calls occur at a rate of 4 calls
per hour (or about 0.07 calls per minute). Statistics tells us that the probability of a call occurring t
minutes after 3:00 AM is given by P  t   1  e0.07t . Determine the likelihood that a person will call
within 5 minutes and 20 minutes of 3:00 AM.
Algebra is:
the study of how to perform multi-step arithmetic calculations more efficiently,
and the study of how to find the correct number to put into a multi-step calculation to get a desired answer.
Int. Alg. Notes
Section 9.2
Page 6 of 6
Applications of Exponential Functions: Radioactive Decay
t
 1  t1/ 2
The amount A of a radioactive isotope left after an amount of time t is given by the formula A  t   A0   ,
 2
where A0 is the initial amount of the isotope, and t1/2 is the half-life of the isotope (i.e., the amount of time for
half of the isotope to decay away).
Practice:
16. Plutonium-239 has a half-life of 24,360 years. Compute how much of a 1,000g sample of Plutonium239 will remain after 100,000 years.
Applications of Exponential Functions: Compound Interest Formula
A principal amount of money P invested in an account with an interest rate of r and which pays compound
interest n times per year will be worth an amount of money A after t years of compounding according to the
nt
 r
formula A  t   P 1   .
 n
Practice:
17. Suppose you invest $1,000.00 in a Roth IRA, and the account pays 8% interest that is compounded
quarterly. Find the value of the investment after 1, 10, and 35 years.
Algebra is:
the study of how to perform multi-step arithmetic calculations more efficiently,
and the study of how to find the correct number to put into a multi-step calculation to get a desired answer.