Chapter 2
Nonlinear Models
Sections 2.1, 2.2, and 2.3
Nonlinear Models
 Quadratic Functions and Models
 Exponential Functions and Models
 Logarithmic Functions and Models
Quadratic Function
A quadratic function of the variable x is a
function that can be written in the form
f ( x)  ax  bx  c
2
 a  0
where a, b, and c are fixed numbers
Example:
f ( x)  12 x  3 x  1
2
Quadratic Function
The graph of a quadratic function is a parabola.
f ( x)  ax  bx  c
2
a>0
 a  0
a<0
Vertex, Intercepts, Symmetry
Vertex coordinates are:
b
x ,
2a
 b 
y  f  
 2a 
y – intercept is:
x0
yc
x – intercepts are solutions of
ax  bx  c  0
2
symmetry
b
x
2a
Graph of a Quadratic Function
Example 1: Sketch the graph of f ( x )  x  2 x  8
2
Vertex:
b
2
x
   1 y  f (1)  9
2a
2
y – intercept
x0
y  8
x – intercepts
x  2x  8  0
2
x  4, 2
Graph of a Quadratic Function
Example 2: Sketch the graph of f ( x)  4 x  12 x  9
2
Vertex:
b
12 3
x


y  f (3 / 2)  0
2a
2 4 2
y – intercept
x0
y 9
x – intercepts
4 x 12 x  9  0 x  3/ 2
2
Graph of a Quadratic Function
1 2
Example 3: Sketch the graph of g ( x)   x  4 x  12
2
Vertex:
b
x
 4 y  f (4)  4
2a
y – intercept
x0
y  12
x – intercepts
1 2
 x  4 x  12  0
2
no solutions
Applications
Example: For the demand equation below, express
the total revenue R as a function of the price p per item
and determine the price that maximizes total revenue.
q( p)  3 p  600
R( p)  pq  p  3 p  600
 3 p  600 p
2
Maximum is at the vertex, p = $100
Applications
Example: As the operator of Workout Fever health Club,
you calculate your demand equation to be q 0.06p + 84
where q is the number of members in the club and p is the
annual membership fee you charge.
1. Your annual operating costs are a fixed cost of $20,000 per
year plus a variable cost of $20 per member. Find the annual
revenue and profit as functions of the membership price p.
2. At what price should you set the membership fee to obtain the
maximum revenue? What is the maximum possible revenue?
3. At what price should you set the membership fee to obtain the
maximum profit? What is the maximum possible profit? What is
the corresponding revenue?
Solution
The annual revenue is given by
R( p)  pq  p  0.06 p  84
 0.06 p  84 p
2
The annual cost as function of q is given by
C (q )  20000  20q
The annual cost as function of p is given by
C ( p )  20000  20q  20000  20  0.06 p  84 
 1.2 p  21680
Solution
Thus the annual profit function is given by
P( p )  R  C  ( 0.06 p 2  84 p )   1.2 p  21680 
 0.06 p 2  85.2 p  21680
The graph of the revenue function R  0.06 p 2  84 p is
b
84
Maximum is at the vertex p  

 $700
2a
2( 0.06)
The graph of the revenue function R  0.06 p 2  84 p is
Maximum revenue is R(700)  $29, 400
The profit function is P( p)  0.06 p 2  85.2 p  21680
b
85.2
Maximum is at the vertex p  

 $710
2a
2(0.06)
The profit function is P( p)  0.06 p 2  85.2 p  21680
Maximum profit is
P(710)  $8, 566
Corresponding Revenue is R(710)  $29,394
Nonlinear Models
 Quadratic Functions and Models
 Exponential Functions and Models
 Logarithmic Functions and Models
Exponential Functions
An exponential function with (constant) base b
and exponent x is defined by
f ( x)  b
x
b  0 and b  1
Notice that the exponent x can be any real number but
the output y= bx is always a positive number. That is,
y  b >0 for all    x  
x
Exponential Functions
We will consider the more general exponential
function defined by
f ( x)  Ab
x
b  0 and b  1
where A is an arbitrary but constant real number.
Example:
f ( x)  5  3
x
Graph of Exponential Functions
when b > 1
f ( x )  Ab x
Graph of Exponential Functions
when 0 < b < 1
f ( x )  Ab x
Graph of Exponential Functions
when b > 1
y2
x
x
y
-4
1/16
-3
1/8
-2
1/4
-1
1/2
0
1
1
2
2
4
3
8
Graphing Exponential Functions
1
y 
2
x
x
y
-3
8
-2
4
-1
2
0
1
1
1/2
2
1/4
3
1/8
4
1/16
Graphing Exponential Functions
y  10
x
y  3x
y  2x
y  1.2
y 1
x
x
Laws of Exponents
Law
Example
1. b x  b y  b x  y
bx
x y
2. y  b
b
 
3. b
x
y
b
xy
4.  ab   a b
x
x x
x
x
a
a
5.    x
b
b
2 2  2  2  8
12
5
123
9
5
5
3
5
6
1
1/ 3
6 / 3
2
8
8
8 
64
3
3 3
3
 2m  2 m  8m
1/ 2
5/ 2
6/ 2
 
1/ 3
 8 
 
 27 
81/ 3
2
 1/ 3 
3
27
3
Finding the Exponential Curve
Through Two Points
Example: Find an exponential curve y Abx that
passes through (1,10) and (3,40).
4b
b2
b  2
2
10  Ab
3
40  Ab
1
Plugging in b2
we get A5
40 Ab

10 Ab
3
f ( x)  5  2
x
Exponential Functions-Examples
A certain bacteria culture grows according to the
following exponential growth model. If the bacteria
numbered 20 originally, find the number of bacteria
present after 6 hours.

Q(6)  20  4
Q(t )  20 40.4479t
When t6

0.4479(6)
  829.86
Thus, after 6 hours there are about 830 bacteria
Compound Interest
r

A(t )  P 1  
 m
mt
A = the future value
P = Present value
r = Annual interest rate (in decimal form)
m = Number of times/year interest is compounded
t = Number of years
Compound Interest
Find the accumulated amount of money after 5
years if $4300 is invested at 6% per year and
interest is reinvested each month
r

A  P 1  
 m
mt
12(5)
 .06 
A  4300 1 

 12 
= $5800.06
The Number e
The exponential function with base e is called
“The Natural Exponential Function”
y  f ( x)  e
x
where e is an irrational constant whose value
is
e  2.718281828459045...
The Natural Exponential Function
The Number e
A way of seeing where the number e comes from,
consider the following example:
If $1 is invested in an account for 1 year at 100%
interest compounded continuously (meaning that
m gets very large) then A converges to e:
m
1

A  1    e
 m
Continuous Compound Interest
A  Pe
rt
A = Future value or Accumulated amount
P = Present value
r = Annual interest rate (in decimal form)
t = Number of years
Continuous Compound Interest
Example: Find the accumulated amount of
money after 25 years if $7500 is invested at
12% per year compounded continuously.
A  Pe
rt
A  7500e
0.12(25)
 $150, 641.53
Exponential Regression
Example: Human population The table shows data for
the population of the world in the 20th century. The
figure shows the corresponding scatter plot.
Exponential Regression
The pattern of the data points suggests exponential growth.
Therefore we try to find an exponential regression model
of the form P(t) Abt
Exponential Regression
We use a graphing calculator with exponential regression
capability to apply the method of least squares and obtain
the exponential model p  (0.008079266)  (1.013731)t
Nonlinear Models
 Quadratic Functions and Models
 Exponential Functions and Models
 Logarithmic Functions and Models
A New Function
How long will it take a $800 investment to be
worth $1000 if it is continuously compounded at
7% per year?
1000  800e
0.07t
5
0.07 t
e
4
Input
Output
A good guess for t is t  3.187765
A New Function
Basically, we take the exponential function with
base b and exponent x,
yb
x
and interchange the role of the variables to define
a new equation
xb
y
 x  0
This new equation defines a new function.
Graphing The New Function
Example: graph the function x 2y
x2
y
x
y
1/16
1/16
-4
1/8
1/8
-3
1/4
1/4
-2
1/2
1/2
-1
1
1
0
2
2
1
4
4
2
8
8
3
Logarithms
The logarithm of x to the base b is the power to
which we need to raise b in order to get x.
y  logb x if and only if x  b
Example: log 3 81 
log 7 1 
log1/ 3 9 
log 5 5 
y
 x  0
Answer: log 3 81  4
log 7 1  0
log1/ 3 9  2
log 5 5  1
Graphing y  log2 x
Recall that y  log2 x is equivalent to x 2y
x2
y
x
y
1/16
1/16
-4
1/8
1/8
-3
1/4
1/4
-2
1/2
1/2
-1
1
1
0
2
2
1
4
4
2
8
8
3
Logarithms on a Calculator
Abbreviations
log x  log10 x Common Logarithm
Natural Logarithm
Base e ln x  log e x
Base 10
log 4  0.60206
ln 26  3.2581
Change of Base Formula
To compute logarithms other than common and
natural logarithms we can use:
log a ln a
logb a 

log b ln b
log15
Example: log9 15 
 1.232487
log 9
Graphs of Logarithmic Function
Properties of Logarithms
1. l og b mn  log b m  log b n
m
2. log b    log b m  log b n
n
3. logb m  n logb m
n
4. log b 1  0
5. logb b  1
Application
Example: How long will it take an $800 investment
to be worth $1000 if it is continuously compounded
at 7% per year?
1000  800e
0.07t
5
 e0.07t
4
Apply ln to both sides
5
ln    0.07t
4
t  3.187765
About 3.2 years
Logarithmic Functions
A more general logarithmic function has the
form
f ( x )  logb x  C
or, alternatively,
f ( x)  A ln x  C
Example:
f ( x)  4.6 ln x  8