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Unformatted text preview: x-axis. Furthermore, as we have mentioned earlier in the text, without
Calculus, the values of the relative maximum and minimum can only be found approximately using
a calculator. If we took the time to ﬁnd the leading term of f , we would ﬁnd it to be x8 . Looking
at the end behavior of f , we notice it matches the end behavior of y = x8 . This is no accident, as
we ﬁnd out in the next theorem.
Theorem 3.2. End Behavior for Polynomial Functions: The end behavior of a polynomial
f (x) = an xn + an−1 xn−1 + . . . + a2 x2 + a1 x + a0 with an = 0 matches the end behavior of y = an xn .
To see why Theorem 3.2 is true, let’s ﬁrst look at a speciﬁc example. Consider f (x) = 4x3 − x + 5.
If we wish to examine end behavior, we look to see the behavior of f as x → ±∞. Since we’re
concerned with x’s far down the x-axis, we are far away from x = 0 and so can rewrite f (x) for
these values of x as 3.1 Graphs of Polynomials 187 f (x) = 4 x3 1 − 5
4x As x becomes unbounded (in either direction), the terms
0, as the table below indicates.
4x2 and 5
4x3 become closer a...
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