Polynomial Functions of Higher Degree Sec. 3.2 Polynomial Functions of Higher Degree

Graphs of Polynomials The graph of a polynomial is continuous (no breaks) p.270 And has only smooth rounded turns (no points like abs. value) p. 270

The simplest is a monomial f(x) = xn n is an integer > 0

Ends of the graph If n is even the ends of the graph go the same direction If n is odd the ends of the graph go different directions The greater n the flatter the graph at the origin P. 271

Ex. What would the ends of the following look like? f(x) = -x3 + 2x2 – 1 f(x) = 3x6 +2x2 …

Zeros of Polynomial Functions Zero – same as x-intercepts or solution Has at most n zeros (n is the degree) Has at most n-1 turning points

Find zeros by factoring Ex. 4 f(x) = x3 – x2 -2x Ex. 5 f(x) = -2x4 + 2x2

If the degree of the factor is even, the graph touches the x-axis but does not cross. If the degree of the factor is odd, the graph crosses the x-axis

Examples f(x) = -x5 + 6x3 -9x f(x) = 3x4 +3x3 – 90x2

Intermediate Value Theorem Let a and b be real numbers such that a < b. If f is a polynomial function such that f(a) ≠ f(b), then, in the interval [a,b], f takes on every value between f(a) and f(b). Tells you there is no break (gap) between two x-values if it is a graph of a polynomial.

Homework P. 280 1-8, 13-19 odd, 28-46 even, 67-77 odd