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

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

Leading Coefficient: coefficient in front of the term with the highest degree It 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. 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 zero mult. 1 factor (x – r) (flattens out) (tangent to axis) Ex 2) Determine an equation (Degree 6) rr r is a zero mult. 3 factor (x – r) 3 r is a zero mult. 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 / Neither Remember: 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) evenodd 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