SOLUTION: Make up a rational function that has the following characteristics: crosses the x-axis at 3; touches the x-axis at -2; has a vertical asymptote at x= 1 and at x= -4; has a hole at

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Question 81824: Make up a rational function that has the following characteristics: crosses the x-axis at 3; touches the x-axis at -2; has a vertical asymptote at x= 1 and at x= -4; has a hole at x=5; has a horizontal asymptote at y= 2.
Please do try to help me! If you can't figure out the whole thing, then it would be a HUGE help in itself if you could tell me what "touches the x-axis at -2" means. What is the difference between that and crossing the x-axis?
Answer by stanbon(75887) (Show Source):
You can put this solution on YOUR website!
Make up a rational function that has the following characteristics:
crosses the x-axis at 3;
Has (x-3) in the numerator.
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touches the x-axis at -2;
Has (x+2)^2 in the numerator.
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has a vertical asymptote at x= 1 and at x= -4;
Has (x-1) and (x+4) in the denominator
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has a hole at x=5;
Has (x-5) in the numerator and in the denominator.
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has a horizontal asymptote at y= 2.
At this point the degrees of the numerator is 4
and the degree of the denominator is 3
so you need a factor of "2" in the numerator and a factor of "x" in the
denominator.
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f(x) = [2(x-3)(x+2)^2(x-5)]/[x(x-1)(x+4)(x-5)]
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What does "touches the x-axis" mean?
Example y=x is a line that "passes thru" the x-axis at x=0.
y=x^2 is a parabola that "touches" the x-axis at x=0
This is what "touches" looks like:

Cheers,
Stan H.