Determine the zeroes, x-intercept, y-intercept, horizontal asymptote, and vertical asymptote.
f(x) = [tex]\frac{5}{x-4}[/tex]
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Determine the zeroes, x-intercept, y-intercept, horizontal asymptote, and vertical asymptote.
f(x) = [tex]\frac{5}{x-4}[/tex]
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Answer:
While vertical asymptotes describe the behavior of a graph as the output gets very large or very small, horizontal asymptotes help describe the behavior of a graph as the input gets very large or very small. Recall that a polynomial’s end behavior will mirror that of the leading term. Likewise, a rational function’s end behavior will mirror that of the ratio of the leading terms of the numerator and denominator functions.
There are three distinct outcomes when checking for horizontal asymptotes:
Case 1: If the degree of the denominator > degree of the numerator, there is a horizontal asymptote at
y
=
0
.
Example:
f
(
x
)
=
4
x
+
2
x
2
+
4
x
−
5
In this case the end behavior is
f
(
x
)
≈
4
x
x
2
=
4
x
. This tells us that, as the inputs increase or decrease without bound, this function will behave similarly to the function
g
(
x
)
=
4
x
, and the outputs will approach zero, resulting in a horizontal asymptote at
y
=
0
. Note that this graph crosses the horizontal asymptote.