Nonlinear Functions
and their Graphs
Lesson 4.1
Polynomials

General formula
P( x)  an x  an 1 x
n




n 1
 ...  a1 x  a0
a0, a1, … ,an are constant coefficients
n is the degree of the polynomial
Standard form is for descending powers of x
anxn is said to be the “leading term”
Extrema of Nonlinear
Functions

Given the function for the Y= screen
y1(x) = 0.1(x3 – 9x2)



Use window -10 < x < 10 and -20 < y < 20
Note the "top of
the hill" and the
"bottom of the
valley"
These are local
extrema
•
•
Extrema of Nonlinear
Functions

Local maximum


f(c) ≥ f(x) when
x is near c
Local minimum

f(n) ≤ f(x) when
x is near n
•
c
n
•
Extrema of Nonlinear
Functions

Absolute minimum

f(c) ≤ f(x) for all x
in the domain of f
•

Absolute maximum

f(c) ≥ f(x) for all x
in the domain of f

Draw a function with an absolute maximum
Extrema of Nonlinear
Functions

The calculator can find maximums and
minimums



When viewing the graph, use the F5 key pulldown
menu
Choose Maximum or Minimum
Specify the upper and lower bound for x (the "near")
Note results
Try It Out

Find local extrema … absolute extrema
1
2
f ( x)   x  1  2
3
h( x)  0.1x5  .02 x 4  .35x3  .36 x 2  1.8x
6
g ( x)  2
x  2x  2
Assignment



Lesson 4.1A
Page 256
Exercises 1 – 45 odd
Even and Odd Functions


If f(x) = f(-x) the graph is symmetric across the
y-axis
It is also an even function
Even and Odd Functions


If f(x) = -f(x) the graph is symmetric across the
x-axis
But ... is it a function ??
Even and Odd Functions

A graph can be symmetric about a point



Called point symmetry
If f(-x) = -f(x) it is symmetric about the origin
Also an odd function
Applications



Consider the U.S. birthrate from 1900 to 2005
(births per 1000 people)
Can be modeled by
3
2
f ( x)  .0000285 x  .0057 x  .48 x  34.4
where x = number of years since 1900
Evaluate f(95)


What does it mean?
With domain 1900 ≤ x ≤ 2005

Identify the absolute minimum and maximum
Applications

U.S. natural gas consumption from 1965 to
1980 can be modeled by
f ( x)  .0001234 x  .005689 x  .08792x  .5145x  1.514
4




3
2
x = 6 is 1966 and x = 20 is 1980
Consumption measured in trillion cubic feet
Evaluate f(10) …. What does it mean?
Graph for 6 ≤ x ≤ 20 and 0.4 ≤ y ≤ 0.8

Determine local extrema, interpret results
Assignment



Lesson 4.1B
Page 258
Exercises 91 – 97 odd