7.2 Graphs of Polynomial Functions

*Quick note: For most of these questions we will use our graphing calculators to graph. A few say “without a graphing utility.” This is when you graph by hand.

Basic Polynomial Functions

Quadratic Cubic Quartic Quintic

Polynomial functions are sums, differences, products or translations of these basic functions

2y x 3y x 4y x 5y x

Relative Extrema: points on a graph that are relative minimums or maximums of the points close to them (like a turning point)

The most a polynomial can have is one less than its degree.

Examples (# of relative extrema):

4 2 none

5 3( ) 5 4f x x x x 5 4 3( ) 8 18 27f x x x x x 5( )f x x

Leading Coefficient: coefficient in front of the term with the highest degreeIt determines if a polynomial rises or falls at the extremes

n is even

a is (+): both up

a is (–): both down

Ex 1) zeros at –1, 0, 2

factors are (x + 1)(x – 0)(x – 2)

We can identify the zeros / roots of a polynomial graph. If we know this, we can find factors and therefore, an equation.

( ) nf x ax

3 2( ) ( 1)( 2) 2f x x x x x x x

n is odd

a is (+): right up, left down

a is (–): right down, left up

* Sometimes polynomials don’t simply pass through the x-axis. If it behaves differently, it means it may be a root with multiplicity.

r

r is a zeromult. 1

factor (x – r)

(flattens out) (tangent to axis)

Ex 2) Determine an equation (Degree 6)

r r

r is a zeromult. 3

factor (x – r)3

r is a zeromult. 2

factor (x – r)2

–6 –3 1 7 down (–) in front

roots: –6, –3, 1 (mult 3), 7

f (x) = –(x + 6)(x + 3)(x – 1)3(x – 7)

Odd / Even / NeitherRemember: If f (–x) = f (x), even function & symmetric wrt y-axis

If f (–x) = – f (x),odd function & symmetric wrt origin

Ex 3) Determine by graphing if polynomial is odd, even, or neither

a)

even

4 2( ) 4 3f x x x 5 3b) ( )f x x x

4a) y x 4b) ( 1) 2y x

odd

Sketching Quickly

Remember horizontal & vertical shifts, & ‘a’ being (+) or (–)

Ex 4) Sketch quickly without graphing calculator

left 1, down 2

Homework

#702 Pg 340 #1–37 odd, 40, 42, 48–51