4.2 Polynomial Functions and Models. A polynomial function is a function of the form.

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Presentation transcript:

4.2 Polynomial Functions and Models

A polynomial function is a function of the form

(a) fxxx()  Determine which of the following are polynomials. For those that are, state the degree. Polynomial. (b) (c) Not a polynomial.

If f is a polynomial function and r is a real number for which f(r)=0, then r is called a (real) zero of f, or root of f. If r is a (real) zero of f, then (a) r is an x-intercept of the graph of f. (b) (x-r) is a factor of f.

Use the above to conclude that x = -1 and x = 4 are the real roots (zeroes) of f.

1 is a zero of multiplicity is a zero of multiplicity is a zero of multiplicity 5.

. If r is a Zero or Odd Multiplicity If r is a Zero or Even Multiplicity

Theorem If f is a polynomial function of degree n, then f has at most n-1 turning points.

Theorem For large values of x, either positive or negative, the graph of the polynomial resembles the graph of the power function.

For the polynomial (a) Find the x- and y-intercepts of the graph of f. (b) Determine whether the graph crosses or touches the x-axis at each x-intercept. (c) Find the power function that the graph of f resembles for large values of x. (d) Determine the maximum number of turning points on the graph of f.

For the polynomial (e) Use the x-intercepts and test numbers to find the intervals on which the graph of f is above the x-axis and the intervals on which the graph is below the x-axis. (f) Put all the information together, and connect the points with a smooth, continuous curve to obtain the graph of f.

(b) -4 is a zero of multiplicity 1. (crosses) -1 is a zero of multiplicity 2. (touches) 5 is a zero of multiplicity 1. (crosses) (d) At most 3 turning points. (a) The x-intercepts are -4, -1, and 5. y-intercept:

Test number: -5 f (-5) 160 Graph of f: Above x-axis Point on graph: (-5, 160)

Test number: -2 f (-2)-14 Graph of f: Below x-axis Point on graph: (-2, -14) -4 < x <-1

Test number: 0 f (0) -20 Graph of f: Below x-axis Point on graph: (0, -20) -1 < x < 5

Test number: 6 f (6) 490 Graph of f: Above x-axis Point on graph: (6, 490)

(6, 490) (5, 0) (0, -20) (-1, 0) (-2, -14)(-4, 0) (-5, 160)