The graph crosses at the x-axis since the exponent on
x - 2 is the odd number 1.
x = 2, the graph changes sign.
(except at the x-intercept, x = -1).
There is a simple rule to
tell whether the graph of a function crosses the x-axis or stays on the same
side at an x-intercept or asymptote at x = c:
If the exponent, n, of the factor (x-c)n in the function is even,
then the graph stays on the same side of the x-axis at c. If the exponent
is odd, then the graph crosses the x-axis at x=c.
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| A vertical asymptote at x = 2 comes from the factor x - 2. The graph crosses at the x-axis since the exponent on x - 2 is the odd number 1. |
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| This time the graph stays on the same side of the axis on both sides of the asymptote at x = 2, since the exponent on the factor x - 2 is the even number 2. | |
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| Here the graph changes sign at both the asymptote at x = 2 and at the x-intercept at -1 since the factors x + 2 and x+ 1 both have odd exponent 1. | |
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| The graph does not change sign at the x-intercept (x = -1)
since the exponent of x + 1 is even. At the asymptote at x = 2, the graph changes sign. |
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| Here, because both exponents are even, the graph stays above
the x-axis the whole time (except at the x-intercept, x = -1). |
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| Again the even exponents keep the graph on one side of the x-axis at both asymptotes. |