Transcript Graphs of Exponential Functions - math-clix

Section 5.2
Exponential
Functions
and Graphs
Copyright ©2013, 2009, 2006, 2001 Pearson Education, Inc.
Objectives

Graph exponential equations and exponential functions.

Solve applied problems involving exponential functions
and their graphs.
Exponential Function
The function f(x) = ax, where x is a real number, a > 0 and
a  1, is called the exponential function, base a.
The base needs to be positive in order to avoid the
complex numbers that would occur by taking even roots
of negative numbers.
The following are examples of exponential functions:
f (x)  2
x
 1
f (x)   
 2
x
f (x)  (3.57)x
Example
Graph the exponential function y = f (x) = 2x.
Example (continued)
As x increases, y increases without bound. As x
decreases, y decreases getting close to 0; as x g ∞, y g
0.
The x-axis, or the line y = 0,
is a horizontal asymptote. As
the x-inputs decrease, the
curve gets closer and closer
to this line, but does not
cross it.
Example
x
 1
Graph the exponential function y  f (x)    .
 2
x
 1
1 x
 2  x.
Note y  f (x)     2
 2
 
This tells us the graph is
the reflection of the graph
of y = 2x across the y-axis.
Selected points are listed
in the table.
Example (continued)
As x increases, the
function values
decrease, getting closer
and closer to 0. The xaxis, y = 0, is the
horizontal asymptote. As
x decreases, the
function values increase
without bound.
Graphs of Exponential Functions
Observe the following graphs of exponential functions
and look for patterns in them.
Example
Graph y = 2x – 2.
The graph is the graph of y = 2x shifted to right 2 units.
Example
x
 1
y  5  0.5  5     5  2  x.
 2
The graph is a reflection of the graph of y = 2x across
the y-axis, followed by a reflection across the x-axis
and then a shift up 5 units.
Graph y = 5 – 0.5x .
x
Application
The amount of money A that a principal P will grow to
after t years at interest rate r (in decimal form),
compounded n times per year, is given by the formula

A  P 1 

nt
r
 .
n
Example
Suppose that $100,000 is invested at 6.5% interest,
compounded semiannually.
a. Find a function for the amount to which the investment
grows after t years.
b. Graph the function.
c. Find the amount of money in the account at t = 0, 4, 8,
and 10 yr.
d. When will the amount of money in the account reach
$400,000?
Example (continued)
Since P = $100,000, r = 6.5%=0.65, and n = 2, we can
substitute these values and write the following function
0.065 

A t   100, 000  1 


2
2t
 $100, 000 1.0325 
2t
Example (continued)
b) Use the graphing calculator with viewing window
[0, 30, 0, 500,000].
Example (continued)
We can compute function values using function notation
on the home screen of a graphing calculator.
Example (continued)
We can also calculate the values directly on a graphing
calculator by substituting in the expression for A(t):
Example (continued)
d)
Set 100,000(1.0325)2t = 400,000 and solve for t,
which we can do on the
graphing calculator.
Graph the equations
y1 = 100,000(1.0325)2t
y2 = 400,000
Then use the intersect
method to estimate the
first coordinate of the
point of intersection.
Example (continued)
Or graph y1 = 100,000(1.0325)2t – 400,000 and use
the Zero method to
estimate the zero of
the function
coordinate of the
point of intersection.
Regardless of the
method, it takes
about 21.67 years,
or about 21 yr, 8 mo,
and 2 days.
The Number e
e is a very special number in mathematics. Leonard Euler
named this number e. The decimal representation of the
number e does not terminate or repeat; it is an irrational
number that is a constant;
e  2.7182818284…
Example
Find each value of ex, to four decimal places, using the ex
key on a calculator.
a) e3
b) e0.23
c) e2
d) e1
a) e3 ≈ 20.0855
b) e0.23 ≈ 0.7945
c) e0 = 1
d) e1 ≈ 2.7183
Graphs of Exponential Functions, Base e
Example
Graph f(x) = ex and g(x) = e–x.
Use the calculator and enter y1 = ex and y2 = e–x.
Enter numbers for x.
Graphs of Exponential Functions,
Base e - Example (continued)
The graph of g is a reflection of the graph of f across
they-axis.
Example
Graph f(x) = ex + 3.
The graph f(x) = ex + 3 is a translation of the graph
of y = ex left 3 units.
Example
Graph f(x) = e–0.5x.
The graph f(x) = e–0.5x is a horizontal stretching of the
graph of y = ex followed by a reflection across the
y-axis.
Example
Graph f(x) = 1  e2x.
The graph f(x) = 1  e2x is a horizontal
shrinking of the graph of y = ex
followed by a reflection across
the y-axis and then
across the x-axis,
followed by a
translation up
1 unit.