GP3-HW11 College Algebra Sketch the graph of each rational function.
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Step 1: Factor the numerator and the denominator. Find the domain.
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{ } Step 2: Rewrite in lowest terms.
The rational function ( )
is in lowest terms.
Step 3: Find the y-intercept, if one exists. Find the x-intercept(s), if any. Plot the points on the graph.
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Step 4: Find the vertical asymptote(s), if any. Graph each vertical asymptote using a dashed line.
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The vertical asymptote is the line .
Step 5: Find the horizontal asymptote or the oblique asymptote, if one exists. Determine the points, if any, at which the graph intersects this asymptote. Graph the asymptote using a dashed line. Plot any points at which the graph intersects the asymptote.
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The degree of the numerator is equal to the degree of the denominator. The horizontal asymptote is the ratio of the leading coefficients. Thus, the horizontal asymptote is the line .
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Step 6: Use the zeros of the numerator and denominator to divide the x-axis into intervals. Determine where the graph is above or below the x-axis by choosing a number in each interval and evaluating the output of the function there. Plot the points found.
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Step 7: Analyze the behavior of the graph near each asymptote indicate this behavior on the graph.
Step 8: Using the results obtained in Steps 1 thru 7 sketch the graph of the rational function.
2.) ( )
Step 1: Factor the numerator and the denominator. Find the domain.
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{ } Step 2: Rewrite in lowest terms.
The rational function ( )
is in lowest terms.
Step 3: Find the y-intercept, if one exists. Find the x-intercept(s), if any. Plot the points on the graph.
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Step 4: Find the vertical asymptote(s), if any. Graph each vertical asymptote using a dashed line.
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Step 5: Find the horizontal asymptote or the oblique asymptote, if one exists. Determine the points, if any, at which the graph intersects this asymptote. Graph the asymptote using a dashed line. Plot any points at which the graph intersects the asymptote.
( )
The degree of the numerator is less than the degree of the denominator, so the horizontal asymptote is the line . We have shown in step 3 that the x-intercepts are , so this rational function intersects its horizontal asymptote twice.
Step 6: Use the zeros of the numerator and denominator to divide the x-axis into intervals. Determine where the graph is above or below the x-axis by choosing a number in each interval and evaluating the output of the function there. Plot the points found.
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Step 7: Analyze the behavior of the graph near each asymptote indicate this behavior on the graph.
Step 8: Using the results obtained in Steps 1 thru 7 sketch the graph of the rational function.
3.) ( )
Step 1: Factor the numerator and the denominator. Find the domain.
( )
( )( )
Step 2: Rewrite in lowest terms.
The rational function ( )
is in lowest terms.
Step 3: Find the y-intercept, if one exists. Find the x-intercept(s), if any. Plot the points on the graph.
( ) ( )
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Step 4: Find the vertical asymptote(s), if any. Graph each vertical asymptote using a dashed line.
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Step 5: Find the horizontal asymptote or the oblique asymptote, if one exists. Determine the points, if any, at which the graph intersects this asymptote. Graph the asymptote using a dashed line. Plot any points at which the graph intersects the asymptote.
( )
The degree of the numerator is greater than the degree of the denominator. It is top heavy by exactly one. There is no horizontal asymptote. There is an oblique asymptote.
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Step 6: Use the zeros of the numerator and denominator to divide the x-axis into intervals. Determine where the graph is above or below the x-axis by choosing a number in each interval and evaluating the output of the function there. Plot the points found.
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Step 7: Analyze the behavior of the graph near each asymptote indicate this behavior on the graph.
Step 8: Using the results obtained in Steps 1 thru 7 sketch the graph of the rational function.
4.) ( )
Step 1: Factor the numerator and the denominator. Find the domain.
( )
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( )( )
{ } Step 2: Rewrite in lowest terms.
The rational function ( )
is in lowest terms.
Step 3: Find the y-intercept, if one exists. Find the x-intercept(s), if any. Plot the points on the graph.
( )
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Step 4: Find the vertical asymptote(s), if any. Graph each vertical asymptote using a dashed line.
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Step 5: Find the horizontal asymptote or the oblique asymptote, if one exists. Determine the points, if any, at which the graph intersects this asymptote. Graph the asymptote using a dashed line. Plot any points at which the graph intersects the asymptote.
( )
The degree of the numerator is greater than the degree of the denominator, so there is no horizontal asymptote. Since its “top heavy” by exactly one, there is an oblique asymptote.
( )
The oblique asymptote is the line
Step 6: Use the zeros of the numerator and denominator to divide the x-axis into intervals. Determine where the graph is above or below the x-axis by choosing a number in each interval and evaluating the output of the function there. Plot the points found.
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Step 7: Analyze the behavior of the graph near each asymptote indicate this behavior on the graph.
Step 8: Using the results obtained in Steps 1 thru 7 sketch the graph of the rational function.
