document.write( "Question 1160127: Write an equation for a rational function with:\r
\n" ); document.write( "\n" ); document.write( "Vertical asymptotes at x = -6 and x = -4\r
\n" ); document.write( "\n" ); document.write( "x intercepts at x = -1 and x = -3\r
\n" ); document.write( "\n" ); document.write( "Horizontal asymptote at y = 7 \n" ); document.write( "
Algebra.Com's Answer #783283 by greenestamps(13200)\"\" \"About 
You can put this solution on YOUR website!


\n" ); document.write( "All the parts of this problem are straightforward, and the reasoning makes sense.

\n" ); document.write( "Memorize the rules about what makes x-intercepts and vertical asymptotes; but also understand why the rules are what they are.

\n" ); document.write( "An x-intercept means the graph crosses (or touches) the x-axis. That means the y value, which is the function value, is 0. In a rational function, the function value is zero whenever the numerator is 0 (unless the denominator is also zero for the same x value).

\n" ); document.write( "This function has x-intercepts at x=-1 and x=-3. That means the numerator of the rational function must contain factors of x+1 and x+3. (The factor x+1 means the numerator will be 0 when x = -1; the factor x+3 means the numerator will be 0 when x = -3.)

\n" ); document.write( "A vertical asymptote means the function is undefined for that value of x. In a rational function, that means there is a factor in the denominator that is zero for that value of x.

\n" ); document.write( "This function has vertical asymptotes at x = -6 and x = -4; that means the denominator has factors of (x+6) and (x+4).

\n" ); document.write( "At this point we have satisfied the requirements for the x-intercepts and vertical asymptotes; our function is of the form

\n" ); document.write( "\"%28%28x%2B3%29%28x%2B1%29%29%2F%28%28x%2B4%29%28x%2B6%29%29\"

\n" ); document.write( "A horizontal asymptote means that for very large positive or very large negative values the function approaches a constant value. In the rational function we have at this point, for very large x values the \"x\" terms dominate (the constants become insignificant), and the function value approaches \"x%5E2%2Fx%5E2+=+1\", so the horizontal asymptote is y=1. To get a horizontal asymptote of y=7, we simply add a constant factor of 7 to the numerator.

\n" ); document.write( "The function we are looking for is then

\n" ); document.write( "\"%287%28x%2B3%29%28x%2B1%29%29%2F%28%28x%2B4%29%28x%2B6%29%29\"

\n" ); document.write( "Here is a graph (window -10,2,-5,5) showing the two x-intercepts:

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\n" ); document.write( "And here is another (window -10,2,-300,300) showing the vertical asymptotes at x=-6 and x=-4:

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