College Algebra 9th txtbk

SECTION 4–1 Polynomial Functions, Division, and Models 263
Explore-Discuss 1 suggests that graphs of polynomial functions with real coefficients
have the properties listed in Theorem 1, which we accept now without proof. Property 3 is
proved later in this section. The other properties are established in calculus.
Z THEOREM 1 Properties of Graphs of Polynomial Functions
Let P(x) be a polynomial of degree n 0 with real coefficients. Then the graph
of P(x):
1. Is continuous for all real numbers
2. Has no sharp corners
3. Has at most n real zeros
4. Has at most n 1 turning points
5. Increases or decreases without bound as x → and as x → *
Figure 5 shows graphs of representative polynomial functions of degrees 1 through 6, illustrating
the five properties of Theorem 1.
y
y
y
5
5
5
5
5
x
5
5
x
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5
x
5
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5
(a) f(x) x 2
(b) g(x) x 3 5x (c) h(x) x 5 6x 3 8x 1
y
y
y
5
5
5
5
5
x
5
5
x
5
5
x
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(d) F(x) x 2 x 1 (e) G(x) 2x 4 7x 2 x 3 (f) H(x) x 6 7x 4 12x 2 x 2
Z Figure 5 Graphs of polynomial functions.
*Remember that and are not real numbers. The statement the graph of P(x) increases without bound as
x → means that for any horizontal line y b there is some interval (, a] {x x a} on which the
graph of P(x) is above the horizontal line.