Nonlinear Functions and their Graphs

Lesson 4.1

Polynomials

General formula

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”

11 1 0( ) ...n n

n nP x a x a x a x a

Extrema of Nonlinear Functions

Given the function for the Y= screeny1(x) = 0.1(x3 – 9x2) Use window -10 < x < 10 and -20 < y < 20

Note the "top of the hill" and the"bottom of thevalley"

These are localextrema

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Extrema of Nonlinear Functions

Local maximum f(c) ≥ f(x) when

x is near c Local minimum

f(n) ≤ f(x) whenx is near n

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cn

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

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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 resultsNote results

Try It Out

Find local extrema … absolute extrema

21( ) 1 2

3f x x

5 4 3 2( ) 0.1 .02 .35 .36 1.8h x x x x x x

2

6( )

2 2g x

x x

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

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

3 2( ) .0000285 .0057 .48 34.4f x x x x

Applications

U.S. natural gas consumption from 1965 to 1980 can be modeled by

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

4 3 2( ) .0001234 .005689 .08792 .5145 1.514f x x x x x

Assignment

Lesson 4.1B Page 258 Exercises 91 – 97 odd