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