Download PreCalc Ch4.1 - LCMR School District

Pre-Calculus Section 4.1: Exponential Functions
Objective: SWBAT evaluate and graph exponential functions and calculate compound
interest.
Homework: Page 336 Day 1 (1-13 odd,19-29 odd)
Day 2 (31-37 odd, 51, 59, 69, 75, 77)
Daily Warm Up:
1.
Solve the system by graphing.
2y  x  2
x  2y  8
Exponential Functions
The exponential function with base a is defined for all real numbers x by
f ( x)  a x
where a > 0 and a  1 .
*We assume a  1 because the function f ( x)  1x  1 is just a constant function.
Example 1: Evaluating Exponential Functions (calculator)
Let f ( x)  3x and evaluate the following:
(a) f(2)
 2
(b) f   
 3
(c) f ( )
(d) f ( 2)
Example 2: Graphing Exponential Functions by Plotting Points
Draw the graph of each function.
(a) f ( x)  3x
1
(b) g ( x)   
3
x
x
f ( x )  3x
1
g ( x)   
3
x
-3
-2
-1
0
1
2
3
y
Note that the family of exponential functions f ( x)  a x have graphs that all pass through
the point (0, 1) because a 0  1 for a  0 . If 0  a  1 , the exponential function decreases
rapidly. If a > 1, the function increases rapidly.
Graphs of Exponential Functions
The exponential function
f ( x)  a x
(a  0, a  1)
has domain R and range (0, ). The line y = 0 (the x-axis) is a horizontal asymptote of f.
Example 3: Identifying Graphs of Exponential Functions
Find the exponential function f ( x)  a x whose graph is given.
.
Example 4: Transformations of Exponential Functions
Use the graph of f ( x)  2 x to sketch the graph of each function.
(a) g ( x)  1  2 x
(b) h( x)  2 x
(c) k ( x)  2 x 1
Example 5: Comparing Exponential and Power Functions
Compare the rates of growth of the exponential function f ( x)  2 x and the power
function g ( x)  x 2 by drawing the graphs of both functions in the following viewing
rectangles.
(a) [0, 3] by [0, 8]
(b) [0, 6] by [0, 25]
(c) [0, 20] by [0, 1000]
The Natural Exponential Function
The natural exponential function is the exponential function
f ( x)  e x
with base e. It is often referred to as the exponential function.
e  2.71828 …
Since 2  e  3 , the graph of the natural exponential function lies between the graphs of
y  2x and y  3x .
Example 6: Evaluating the Exponential Function
Evaluate each expression correct to five decimal places.
(a) e3
(b) 2e0.53
(c) e 4.8
Example 7: Transformations of the Exponential Function
Sketch the graph of each function.
(a) f ( x)  e x
(b) g ( x)  3e0.5 x
Example 8: An Exponential Model for the Spread of a Virus
An infectious disease begins to spread in a small city of population 10,000. After
t days, the number of persons who have succumbed to the virus is modeled by the
function
10, 000
v(t ) 
5  1245e 0.97 t
(a) How many infected people are there initially (at time t = 0)
(b) Find the number of infected people after one day, two days, and five days.
(c) Use your calculator to graph the function v and describe its behavior.
Compound Interest
Compound Interest is calculated by the formula
nt
 r
A(t )  P 1  
 n
where A(t) = amount after t years
P = principal
r = interest rate per year
n = number of times interest is compounded per year
t = number of years
Example 9: Calculating Compound Interest
A sum of $1000 is invested at an interest rate of 12% per year. Find the amounts
in the account after 3 years if interest is compounded annually, semiannually, quarterly,
monthly, and daily.
Continuously Compounded Interest
Continuously compounded interest is calculated by the formula
A(t )  Pert
where A(t) = amount after t years
P = principal
r = interest rate per year
t = number of years
Example 10: Calculating Continuously Compound Interest
Find the amount after 3 years if $1000 is invested at an interest rate of 12% per
year, compounded continuously.