SFM Productions Presents: Another day of Pre-Calculus torture! No fun for you - tons of fon for me! 2.2 Polynomial Functions of Higher Degree

Homework for section 2.2 p all, 21-29, eoo, eoo, eoo, 97,99

A Polynomial function is continuous. A Polynomial function has only smooth rounded turns. If we take the simplest function, f(x) = x n if n is odd, then the graph goes through the x-axis if n is even, then the graph touches the x-axis (in certain cases, the graph doesn’t even have to touch the x-axis at all…)

f(x)=x 2 or f(x) = x 4 or f(x) = x 6

f(x)=x 3 or f(x) = x 5 or f(x) = x 7

What about more “challenging” functions: f(x)= x 4 + 3x 3 - 2x + 5 and f(x)= -x4 - 3x3 + 2x + 5

The Leading Coefficient test (and what it means) If n is evenIf n is odd If a>0 If a<0

In the function f(x), with some degree n : The function has, at most, n-1 turning points. The function has, at most, n real zeros and speaking of REAL zeros………

REAL Zeros of Polynomial Functions If f(x) is a polynomial function, and a is a real number, the following things all mean the same thing. a is a root of the function f(x) x = a is a zero of the function f(x) x = a is a solution of the polynomial equation f(x) = 0 (x-a) is a factor of the polynomial f(x) (a, 0) is an x-intercept of the graph of f

Sketching the graph of polynomial functions, or, why we’re doing all of this... Sketch f(x) = -2x 4 + 2x 2 First, look at the leading coefficient and degree to determine what the graph is doing at the extreme left and the extreme right. Then, find the zeros of the function… -2x 4 + 2x 2 = 0 X = 0 X = ± 1 -2x 2 (x 2 - 1) = 0 Note that x 2 = 0 means that x = 0 and x = 0. ∴ the repeated zero has a multiplicity of 2 (which is even) because there are two of the same number

A multiplicity that is even means that the graph of the function touches the x-axis at that zero. A multiplicity that is odd means that the graph of the function goes through the x-axis at that zero. In the example from the previous slide, we have a zero of even (2) multiplicity at 0, anda zero of odd (1) multiplicity at -1, anda zero of odd (1) multiplicity at 1. And, since we remember (?) what the graph is doing at the extreme left and extreme right, we can sketch a very reasonable graph of f(x) = -2x 4 + 2x 2

Leading coefficient test tell us… Rises left and Falls right Zeros: Mult = 1 (odd) Mult = 2 (even)

Inflection Point

Finding polynomials when given the zeros… If x = 3 and x = 8, find a polynomial that fits. If x = 3, then 3 is a zero (it is also a solution) Same thing for x = 8. That means that (x - 3) and (x - 8) are factors. (x - 3)(x - 8) = 0 x x + 24 = 0 a(x - 3)(x - 8) = 0 a could be any number, so there are an infinite number of “correct” answers 2x x + 48 = 0 3x x + 72 = 0 All three have a different shape…due to different coefficients…which cause different stretches.

The Intermediate Value Theorem ? Somewhere, this graph has to cross zero…

Go! Do!