Sec 3 2 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

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

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

Ex. What would the ends of the following look like? 1) f(x) = -x 3 + 2 x 2 – 1 2) f(x) = 3 x 6 +2 x 2 …

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) = x 3 – x 2 -2 x • Ex. 5 § f(x) = -2 x 4 + 2 x 2

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 a) f(x) = -x 5 + 6 x 3 -9 x b) f(x) = 3 x 4 +3 x 3 – 90 x 2

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