Chapter 3

Section 3.2

Polynomial Functions and Models

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Polynomials Example

where f(x) is the U.S. consumption of natural gas in trillion cubic feet from 1965 to 1980 and x is the number of years after 1960

Source: U.S. Department of Energy

f(x) is called a polynomial function

The expression for f(x) is called a polynomial

Questions: Is f(x) linear ? What is f(10) ?

Polynomial Functions

f(x) = 0.0001234x4 – 0.005689x3 + 0.08792x2 – 0.5145x + 1.514

What does this mean in terms of the model ?f(10) = 0.706

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Polynomial Functions Polynomial Terminology

Polynomial

anxn + an-1xn-1 + … + a1x + a0

a polynomial of degree n

Polynomial Function

f(x) = anxn + an-1xn-1 + … + a1x + a0

a polynomial function of degree n

Polynomial Equation

anxn + an-1xn-1 + … + a1x + a0 = 0

an nth degree polynomial equation in standard form

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Polynomial Functions Polynomials and Their Graphs

Turning Point

A point where the graph changes from increasing to decreasing or vice versa

Occurs at local minimum or local maximum

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How many turning points ?

How many local extrema ?

How many x-intercepts ?

How many y-intercepts ?

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Polynomial Functions Polynomials By Degree

Zero Degree Constant function: f(x) = a , a ≠ 0 If a = 0 , degree is undefined

(i.e. no degree) Graph: Horizontal line No turning points No x-intercepts

First Degree Linear function: f(x) = ax + b , a ≠ 0 Graph: Non-vertical, non-horizontal line No turning points One x-intercept End behavior: opposite directions

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Polynomial Functions Polynomials By Degree

Second Degree Quadratic function/equation:

f(x) = ax2 + bx + c , a ≠ 0 Graph: Parabola One turning point At most two x-intercepts End behavior: same direction

Third Degree Cubic function/equation:

f(x) = ax3 + bx2 + cx + d , a ≠ 0 Graph: Two or no turning points One to three x-intercepts End behavior: opposite directions

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Polynomial Functions Polynomials By Degree

Fourth Degree Quartic function/equation: f(x) = ax4 + bx3 + cx2 + dx + e , a ≠ 0 Graph: At most three turning points At most four x-intercepts End behavior: same direction

Fifth Degree Quintic function/equation: f(x) = ax5 + bx4 + cx3 + dx2 + ex + k , a ≠ 0 Graph: At most four turning points At most five x-intercepts End behavior: opposite directions

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Polynomials By Degree nth Degree

Degree n function/equation:

f(x) = anxn + an–1x

n–1 + … + a2x2 + a1x + a0 , an ≠ 0

Graph: At most n – 1 turning points

At most n x-intercepts End behavior: depends on a and n

For a > 0 :

For a < 0 : opposite behavior from above

Polynomial Functions

f(x) ∞ if n odd or even

if n odd

if n even

as x ∞as x ∞– f(x) ∞–

as x ∞– f(x) ∞

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2. f(x) = –3x2 + 1

4. f(x) = –x3 + x

Sketch the graphs of the following: 1. f(x) = 3x2 + 1

3. f(x) = x3 – x

Polynomial Function Examples

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Sketch the graphs (continued): 5. f(x) = x4 – 5x2 + 4

7. f(x) = x5 – 5x3 + 4x

6. f(x) = –x4 + 5x2 – 4

8. f(x) = –x5 + 5x3 – 4x

Polynomial Function Examples

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Think about it !