SOLUTION: PLEASE help! My teacher didn't explain. I have to write a rational function with the given asymptotes. a. x=-2, y=0 b. x=4,y=0 c.x=2,x=1,y=1 d.x=0,y= -1 Thank you so muc

Algebra ->  Rational-functions -> SOLUTION: PLEASE help! My teacher didn't explain. I have to write a rational function with the given asymptotes. a. x=-2, y=0 b. x=4,y=0 c.x=2,x=1,y=1 d.x=0,y= -1 Thank you so muc      Log On


   

Question 583086: PLEASE help! My teacher didn't explain.
I have to write a rational function with the given asymptotes.
a. x=-2, y=0
b. x=4,y=0
c.x=2,x=1,y=1
d.x=0,y= -1
Thank you so much!
Found 2 solutions by KMST, solver91311:
Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
a) Vertical asymptotes. like x=-2, happen only when a denominator is zero.
f%28x%29=1%2F%28x%2B2%29 has a vertical asymptote at x=-2
As x approaches -2, the denominator approaches zero and the absolute value of f(x) grows without bounds. The graph of f%28x%29=1%2F%28x%2B2%29 looks like this:
graph%28300%2C300%2C-5%2C1%2C-50%2C50%2C1%2F%28x%2B2%29%29 Coincidentally that function also has y=0 as an asymptote, as you can see from the graph.
As the absolute value of x (and consequently of x+2) grows larger, and larger, f(x) grows closer and closer to zero. A y=0 horizontal asymptote happens when your rational function is a quotient and the denominator polynomial has a higher degree than the numerator.
CAUTION: Not every time a denominator is zero, you have a vertical asymptote.
If you make sure that the denominator, and only the denominator is zero at x=-2, you can be sure that the function will have an x=-2 asymptote.
If the numerator and denominator are zero at the same time, the function can be equivalent to another function that does not have a vertical asymptote.
For example, p%28x%29=%28x%2B2%29%2F%28x%2B2%29 is equivalent to q%28x%29=1 for all values of x except x=-2, and you know that q%28x%29=1 graphs as a horizontal line with y=1, and does not have a vertical asymptote. The graph for p%28x%29=%28x%2B2%29%2F%28x%2B2%29 looks just like the same horizontal y=1 line, except for a hole at x=-2, where p(x) does not exist.
b) From what I said above, you must realize that for a vertical x=4 asymptote, you need the denominator to be zero for x=4.
g%28x%29=1%2F%28x-4%29 would work. It also has a y=0 asymptote, because the denominator, x-4 has degree 1, and the numerator, 1, has degree zero.
c) h%28x%29=1%2F%28%28x-1%29%28x-2%29%29 has asymptotes x=1 and x=2 because those are zeros of the denominator.
The only horizontal asymptote for h%28x%29=1%2F%28%28x-1%29%28x-2%29%29 is y=0 and we need y=1, but that is easy to fix: we just add 1.
m%28x%29=1%2Bh%28x%29=1%2B1%2F%28%28x-1%29%28x-2%29%29 has x=1, x=2 and y=1 asymptotes.
You can make it look fancier:
m%28x%29=1%2B1%2F%28%28x-1%29%28x-2%29%29=%28x-1%29%28x-2%29%2F%28%28x-1%29%28x-2%29%29+1%2F%28%28x-1%29%28x-2%29%29=%28x%5E2-3x%2B2%29%2F%28x%5E2-3x%2B2%29+1%2F%28x%5E2-3x%2B2%29=%28x%5E2-3x%2B3%29%2F%28x%5E2-3x%2B2%29
d) For an x=0 asymptote we want x as a factor in the denominator, but not tin the numerator. A y=0 asymptote would be easier, but you saw in part c) how you can get a horizontal asymptote at a different y value
r%28x%29=-1%2B1%2Fx=%281-x%29%2Fx has x=0 and y=-1 asymptotes.
s%28x%29=-1+1%2Fx%5E2=%281-x%5E2%29%2Fx%5E2 would work too.

Answer by solver91311(24713) About Me  (Show Source):
You can put this solution on YOUR website!


A rational function has a vertical asymptote wherever the function is undefined, that is wherever the denominator is zero. If the denominator is zero only when , then a possible expression for your denominator is since iff . A more general expression that provides the same result is where .

A rational function has a horizontal asymptote of 0 only when the degree of the numerator is strictly less than the degree of the denominator. For your specific case denominator above which has a degree of 1 you must have a numerator of degree zero, which is to say some constant. For the general case, your numerator must have a degree no greater than .

Specific case:



General case:



where at least one

John

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