College Algebra 9th txtbk

264 CHAPTER 4 POLYNOMIAL AND RATIONAL FUNCTIONS
EXAMPLE 2
Properties of Graphs of Polynomials
Explain why each graph is not the graph of a polynomial function by listing the properties
of Theorem 1 that it fails to satisfy.
(A) (B) (C)
y
y
y
5
5
5
5
5
x
5
5
x
5
5
x
5
5
5
SOLUTIONS
(A) The graph has a sharp corner when x 0. Property 2 fails.
(B) There are no points on the graph with x coordinate less then or equal to 0, so
properties 1 and 5 fail.
(C) There are an infinite number of zeros and an infinite number of turning points, so
properties 3 and 4 fail. Furthermore, the graph is bounded by the horizontal lines
y 1, so property 5 fails.
MATCHED PROBLEM 2
Explain why each graph is not the graph of a polynomial function by listing the properties
of Theorem 1 that it fails to satisfy.
(A) (B) (C)
y
y
y
5
5
5
5
5
x
5
5
x
5
5
x
5
5
The shape of the graph of a polynomial function with real coefficients is similar to the
shape of the graph of the leading term, that is, the term of highest degree. Figure 6 compares
the graph of the polynomial h(x) x 5 6x 3 8x 1 from Figure 5 with the graph
of its leading term p(x) x 5 . The graphs are dissimilar near the origin, but as we zoom
out, the shapes of the two graphs become quite similar. The leading term in the polynomial
dominates all other terms combined. Because the graph of p(x) increases without bound as
x → , the same is true of the graph of h(x). And because the graph of p(x) decreases without
bound as x → , the same is true of the graph of h(x).
5
Z Figure 6 p(x) x 5 ,
h(x) x 5 6x 3 8x 1.
5
y
p h
500
y
ph
5
5
x
ZOOM OUT
5
5
x
5
500