document.write( "Question 390783: Make up a rational function graph that meets the following conditions:
\n" ); document.write( "a)it will have an oblique asymptote of slope \"not equal\" to 1.
\n" ); document.write( "b)it will have two x-intercepts, one of multiplicity 1 and the other 2.
\n" ); document.write( "c)it will have two vertical asymptotes.
\n" ); document.write( "Then, go through the complete process of graphing your rational function, showing all steps, including the long division. \n" ); document.write( "

Algebra.Com's Answer #277232 by jsmallt9(3758) $\"\"$ $\"About$

You can put this solution on YOUR website!
First we'll look at part b. The x-intercepts of a rational function will be where x makes the numerator zero. In order for an x-intercept to exist, (x-r) (where r is the intercept) must be a factor of the numerator. And for one x-intercept to be of multiplicity 2, (x-r) for that intercept must be a factor twice. So if $\"r%5B1%5D\"$ is the x-intercept of multiplicity 1 and $\"r%5B2%5D\"$ is the intercept of multiplicity 2, then the numerator will have the following factors:
\n" ); document.write( " $\"%28x-r%5B1%5D%29%28x-r%5B2%5D%29%28x-r%5B2%5D%29\"$ or $\"%28x-r%5B1%5D%29%28x-r%5B2%5D%29%5E2\"$

\n" ); document.write( "Next we'll look at part c. For a rational function to have two vertical asymptotes, there must be two numbers that make the denominator zero. If we call these two numbers $\"r%5B3%5D\"$ and $\"r%5B4%5D\"$ , then the denominator must have the factors:
\n" ); document.write( " $\"%28x-r%5B3%5D%29%28x-r%5B4%5D%29\"$

\n" ); document.write( "Last of all we will look at part a. So far our function looks like:
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\n" ); document.write( "I hope it is easy to see that if we were to multiply out the numerator and denominator, the highest power term in the numerator would be $\"x%5E3\"$ and the highest power term in the denominator would be $\"x%5E2\"$ . And if we divide $\"x%5E3\"$ by $\"x%5E2\"$ we would get just \"x\". This would give us an oblique asymptote with a slope of 1 (since the coefficient of x is the slope). But we do not want a slope of 1. To get a different slope we need for the coefficients of the highest power terms to be different. One way to get different coefficients would be to add a constant factor to the numerator or denominator that is not a 1. If you add the factor to the numerator then that number will become the slope of the oblique asymptote.

\n" ); document.write( "So our function now looks like:
\n" ); document.write( "

\n" ); document.write( "All we need to do now is pick numbers for \"a\", $\"r%5B1%5D\"$ , $\"r%5B2%5D\"$ , $\"r%5B3%5D\"$ and $\"r%5B4%5D\"$ . In order to get the function you want, pick any number but 1 for \"a\" and make sure that $\"r%5B1%5D\"$ , $\"r%5B2%5D\"$ , $\"r%5B3%5D\"$ and $\"r%5B4%5D\"$ are all different numbers. As long as you follow the above rules your function will meet all the requirements of the problem no matter what numbers you pick!

\n" ); document.write( "As for the graphing, here are some key ideas:

The vertical asymptotes will be $\"x+=+r%5B3%5D\"$ and $\"x+=+r%5B4%5D\"$ .
The graph will cross the x-axis at $\"r%5B1%5D\"$
The graph will have a relative maximum or minimum at the point ( $\"r%5B4%5D\"$ , 0). (If you do not know what a \"relative maximum or minimum\" means, then the graph will \"bounce off\" the x-axis at ( $\"r%5B4%5D\"$ , 0)). (This happens because of the multiplicity of 2.)
If x=0 is not a vertical asymptote, then the graph will have a y-intercept. Put a zero in for x and figure out the value of the function when x is zero to determine the y-intercept.
To find the equation of the oblique asymptote:
1. Multiply out both the numerator and the denominator.
2. Use long division to divide the numerator by the denominator.
3. You should get something in the form of:
  \n" ); document.write( "ax + b + expression/denominator
  \n" ); document.write( "For large x values the expression/denominator fraction will approach zero. So the oblique asymptote will be the line:
  \n" ); document.write( "y = ax + b
  \n" ); document.write( "where \"a\" is the number you picked earlier and \"b\" is some other number (maybe even the same as \"a\") depending on how the division works out.
Two x-intercepts, an oblique asymptote and possibly a y-intercept are not enough to figure out the whole graph. More points are needed. Pick some additional values for x and find the y value for each one. This will give you additional points. Keep finding additional points and plot them until you have a sense as to how the graph goes. Among the x values you try, I would suggest picking x values a little to the left and a little to the right of each of the two vertical asymptotes.

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