2.2 Polynomial Functions of Higher Degree

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2.2 Polynomial Functions of Higher Degree - PowerPoint PPT Presentation

2.2 Polynomial Functions of Higher Degree. Note: We will be talking about continuous functions in 3.2. f(x) = ax n where n. If n is odd and a &gt; 0, then f(x) will end in Quad’s I &amp; III. If n is odd and a &lt; 0, then f(x) will end in Quad’s II &amp; IV.

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2.2

Polynomial Functions

of Higher Degree

Note: We will be talking about

continuous functions in 3.2

f(x) = axn where n

If n is odd and a > 0, then f(x) will end in Quad’s I & III

If n is odd and a < 0, then f(x) will end in Quad’s II & IV

If n is even and a > 0, then f(x) will end in Quad’s I & II

If n is even and a < 0, then f(x) will end in Quad’s III & IV

What quadrants will the following functions finish in?

f(x) = -x3 +4x

f(x) = x4 – 5x2 + 4

f(x) = x5 – x

f(x) = -x2 + 4x - 7

II & IV

I & II

I & III

III & IV

A zero of a function f is an x-value for which f(x) = 0.

In other words, zeros are the x-intercepts orsolutions

of the polynomial f(x) = 0.

Ex. Find the real zeros of f(x) = x3 – x2 – 2x

Graph

0 = x3 – x2 – 2x

0 = x(x2 – x – 2)

0 = x(x – 2)(x + 1)

0, 2, -1 are the real zeros.

+ x3 means it ends in what

I, III

Ex. Find the real zeros of f(x) = -2x4 + 2x2

0 = -2x4 + 2x2

0 = -2x2(x2 – 1)

0 = -2x2(x – 1)(x + 1)

zeros 0, 1, -1

a -x4 end up in?

III, IV

Ex. Find a polynomial with the given zeros.

a. -2, -1, 1, 2

f(x) = (x + 2)(x + 1)(x – 1)(x – 2)

Foil these first

f(x) = (x2 – 4)(x2 – 1)

f(x) = x4 – 5x2 + 4

(x + ½) or

(2x + 1)

b. -1/2, 3, 3

f(x) = (2x + 1)(x – 3)(x – 3)

f(x) = (2x + 1)(x2 – 6x + 9)

f(x) = 2x3 – 11x2 + 12x + 9

Use the Intermediate Value Theorem to approximate

the real zero of f(x) = x3 – x2 + 1

Use graphing calculator to graph f(x) and then

use the table to find the zero.