4.1. More Worked Examples: Graphs of Simple Rational Functions |
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Characterize the graph of each of the following by naming a) the domain b) the end behavior: As and as c) the asymptotes d) the intercepts
Use this information to sketch the graph by hand , finding at least one additional point on either side of the vertical asymptote. Then name e) the range and f) the interval(s) over which the function is increasing and/or decreasing. . |
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Example 1. |
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Set the denominator = 0 to find the values for which the function is undefined. The domain is all real numbers EXCEPT this value.
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is the domain |
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Choose some very large values for x and some very small values for x to determine end behavior.
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As As |
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The vertical asymptote(s) occurs at the value(s) for which the function is undefined.
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The vertical asymptote is |
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The end behavior determines the horizontal asymptote. |
The horizontal asymptote is |
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To find the y-intercept, set |
, which is undefined. There is no y-intercept. |
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To find the x-intercept(s) , set
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Thus, there are no x-intercepts
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Finding a few extra points makes our graph more accurate. |
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Plotting the above, we obtain: |
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The graph of |
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Note that the range (the set of all outputs) is the set of all real numbers except 0. The function is decreasing over its entire domain, that is it is decreasing over . |
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Example 2. |
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Set the denominator = 0 to find the values for which the function is undefined. The domain is all real numbers EXCEPT this value.
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is the domain |
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Choose some very large values for x and some very small values for x to determine end behavior.
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As As |
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The vertical asymptote(s) occurs at the value(s) for which the function is undefined.
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The vertical asymptote is |
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The end behavior determines the horizontal asymptote. |
The horizontal asymptote is |
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To find the y-intercept, set |
, which is undefined. There is no y-intercept. |
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To find the x-intercept(s) , set
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Thus, there are no x-intercepts.
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Finding a few extra points makes our graph more accurate. |
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Plotting the above, we obtain:
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The graph of |
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Example 3. |
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Set the denominator = 0 to find the values for which the function is undefined. The domain is all real numbers EXCEPT this value.
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is the domain |
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Choose some very large values for x and some very small values for x to determine end behavior.
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As As |
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The vertical asymptote(s) occurs at the value(s) for which the function is undefined.
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The vertical asymptote is |
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The end behavior determines the horizontal asymptote. |
The horizontal asymptote is |
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To find the y-intercept, set |
, which is undefined. There is no y-intercept. |
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To find the x-intercept(s) , set
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Thus, there are no x-intercepts.
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Finding a few extra points makes our graph more accurate. |
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Plotting the above, we obtain:
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The graph of |
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Example 4. |
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Set the denominator = 0 to find the values for which the function is undefined. The domain is all real numbers EXCEPT this value.
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is the domain |
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Choose some very large values for x and some very small values for x to determine end behavior.
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As As |
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The vertical asymptote(s) occurs at the value(s) for which the function is undefined.
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The vertical asymptote is |
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The end behavior determines the horizontal asymptote. |
The horizontal asymptote is |
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To find the y-intercept, set |
, which is undefined. There is no y-intercept. |
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To find the x-intercept(s) , set
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Thus, there are no x-intercepts.
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Finding a few extra points makes our graph more accurate. |
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Plotting the above, we obtain:
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The graph of |
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