7.2 Graphs of Polynomial Functions

1. 7.2 Graphs of Polynomial Functions

2. *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 Quintic Cubic Quartic Polynomial functions are sums, differences, products or translations of these basic functions

3. 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

4. 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 n is odd a is (+): right up, left down a is (–): right down, left up We can identify the zeros / roots of a polynomial graph. If we know this, we can find factors and therefore, an equation. Ex 1) zeros at –1, 0, 2 factors are (x + 1)(x – 0)(x – 2)

5. * 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 r (tangent to axis) (flattens out) r is a zero mult. 2 factor (x – r)2 r is a zero mult. 1 factor (x – r) r is a zero mult. 3 factor (x – r)3 Ex 2) Determine an equation (Degree 6) roots: –6, –3, 1 (mult 3), 7 • f (x) = –(x + 6) • (x + 3) • (x – 1)3 • (x – 7) –6 7 –3 1 down  (–) in front

6. Odd / Even / Neither Remember: If f (–x) = f (x), even function & symmetric wrty-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 odd Sketching Quickly Remember horizontal & vertical shifts, & ‘a’ being (+) or (–) Ex 4) Sketch quickly withoutgraphingcalculator left 1, down 2

7. Homework #702 Pg 340 #1–37 odd, 40, 42, 48–51