POLYNOMIAL FUNCTIONS AND MODELS SECTION 3.1 POLYNOMIAL FUNCTIONS AND MODELS

f(x) = a n x n + a n-1 x n-1 + . . . + a1x + a0 POLYNOMIAL FUNCTIONS A polynomial is a function of the form f(x) = a n x n + a n-1 x n-1 + . . . + a1x + a0 where an, a n-1, . . ., a1, a0 are real numbers and n is a nonnegative integer. The domain consists of all real numbers.

POLYNOMIAL FUNCTIONS Which of the following are polynomial functions?

POLYNOMIAL FUNCTIONS SEE TABLE 1

POLYNOMIAL FUNCTIONS The graph of every polynomial function is smooth and continuous: no sharp corners and no gaps or holes.

POLYNOMIAL FUNCTIONS When a polynomial function is factored completely, it is easy to solve the equation f(x) = 0 and locate the x-intercepts of the graph. Example: f(x) = (x - 1)2 (x + 3) = 0 The zeros are 1 and - 3

POLYNOMIAL FUNCTIONS 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.

POLYNOMIAL FUNCTIONS If (x - r)m is a factor of a polynomial f and (x - r)m+1 is not a factor of f, then r is called a zero of multiplicity m of f. Example: f(x) = (x - 1)2 (x + 3) = 0 1 is a zero of multiplicity 2.

POLYNOMIAL FUNCTIONS For the polynomial f(x) = 5(x - 2)(x + 3)2(x - 1/2)4 2 is a zero of multiplicity 1 - 3 is a zero of multiplicity 2 1/2 is a zero of multiplicity 4

INVESTIGATING THE ROLE OF MULTIPLICITY For the polynomial f(x) = x2(x - 2) (a) Find the x- and y-intercepts of the graph. (b) Graph the polynomial on your calculator. (c) For each x-intercept, determine whether it is of odd or even multiplicity. What happens at an x-intercept of odd multiplicity vs. even multiplicity?

EVEN MULTIPLICITY If r is of even multiplicity: The sign of f(x) does not change from one side to the other side of r. The graph touches the x-axis at r.

ODD MULTIPLICITY If r is of odd multiplicity: The sign of f(x) changes from one side to the other side of r. The graph crosses the x-axis at r.

TURNING POINTS When the graph of a polynomial function changes from a decreasing interval to an increasing interval (or vice versa), the point at the change is called a local minima (or local maxima). We call these points TURNING POINTS.

EXAMPLE Look at the graph of f(x) = x3 - 2x2 How many turning points do you see? Now graph: y = x3, y = x3 - x, y = x3 + 3x2 + 4

EXAMPLE Now graph: y = x4, y = x4 - (4/3)x3, y = x4 - 2x2 How many turning points do you see on these graphs?

THEOREM If f is a polynomial function of degree n, then f has at most n - 1 turning points. In fact, the number of turning points is either exactly n - 1or less than this by a multiple of 2.

GRAPH: P(x ) = x2 P2(x) = x3 P1(x) = x4 P3(x) = x5

When n (or the exponent) is even, the graph on both ends goes to ± ¥. When n is odd, the graph goes in opposite directions on each end, one toward + ¥ , the other toward - ¥.

EXAMPLE: Determine the direction the arms of the graph should point. Then, confirm your answer by graphing. f(x) = - 0.01x 7

EXAMPLE: Graph the functions below in the same plane, first using [- 10,10] by [- 1000, 1000], then using [- 10, 10] by [- 10000, 10000]: p(x) = x 5 - x 4 - 30x 3 + 80x + 3 p(x) = x 5

The behavior of the graph of a polynomial as x gets large is similar to that of the graph of the leading term.

f(x) = a n x n + a n-1 x n-1 + . . . + a1x + a0 THEOREM For large values of x, either positive or negative, the graph of the polynomial f(x) = a n x n + a n-1 x n-1 + . . . + a1x + a0 resembles the graph of the power function y = a n x n

EXAMPLE DO EXAMPLES 9 AND 10

CONCLUSION OF SECTION 3.1