Unformatted text preview: f (x) touches and rebounds from the x-axis as (c, 0).
• If m is odd, the graph of y = f (x) crosses through the x-axis as (c, 0). 3.1 Graphs of Polynomials 189 Our last example shows how end behavior and multiplicity allow us to sketch a decent graph without
appealing to a sign diagram.
Example 3.1.6. Sketch the graph of f (x) = −3(2x − 1)(x + 1)2 using end behavior and the
multiplicity of its zeros.
Solution. The end behavior of the graph of f will match that of its leading term. To ﬁnd
the leading term, we multiply by the leading terms of each factor to get (−3)(2x)(x)2 = −6x3 .
This tells us the graph will start above the x-axis, in Quadrant II, and ﬁnish below the x-axis, in
Quadrant IV. Next, we ﬁnd the zeros of f . Fortunately for us, f is factored.16 Setting each factor
equal to zero gives is x = 2 and x = −1 as zeros. To ﬁnd the multiplicity of x = 1 we note that
it corresponds to the factor (2x − 1). This isn’t strictly in the form required in Deﬁnition 3.3. If
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