Question

Find equations for the horizontal or oblique asymptotes, if any, for each of the following rational functions.

#1) F(x)=5/x-1
#2)F(x)= x^2-4/x+2
#3) F(x)= x+2/x^2-9
#4) F(x)=3x^2+1/x-2

Can you please show your work so I can understand how you got the answer. Thank you! I will definitely award medal

Answer 1

Do you know the rules for the horiz asyp? You just compare the degree of the num'r with degree of the den'r.

If degree num'r<degree den'r, then horz asymp is the line y=0.. The denominator will "take over" as x gets large, and the function value with tend to 0.

If degree num'r=degree den'r, then horz asymp is the line y=p/q, where p is the leading coefficient of the num'r and q is the leading coeff of the den'r. The leading terms will "take over" as x gets large, and the x^n terms essentially cancel and you are left with the ration of the leading coefficients.

Answer 2

If degree num'r>degree den'r, that's where you get an oblique asymptote.

In that case, you have to do the division, e.g., divide the num'r by the den'r. Whatever you get as the quotient is your equation for the oblique asymptote. You can ignore the remainder, because that part will -->0 as x gets large.

Answer 3

This is my answer for 5/x-1
VA: x=1
HA: y=0
Y-intercept: -5
No x-intercept

Now how would I graph this? Thank you so very much for that help debbie

Answer 4

Good, all of that looks perfect.

Put all of those things on the graph first. Then think about some values near the VA. E.g., to the LEFT of 1 (going towards 1, but from values that are less than 1), what is happening to the function? There's a VA there, so we know it's "blowing up"..... but is it blowing up to +infinity or to -infinity? (Think about the sign of the function when x<1). This will tell you the behavior as it approaches that VA from the left.

Answer 5

Ask the same question as you approach it from the right, e.g., as you get CLOSE to x=1 from from values where x>1.... what is the sign of the function? That tells you "which direction" it is blowing up.

Answer 6

You also have the y-intercept, so of course plot that.

You can find a couple of other points if you want, but really.... if you have a "feel" for what the function looks like, once you have the asymptotes and any intercepts on the graph, you can get a pretty good sketch.

Answer 7

Also if you happen to be familiar with the graph of \(y=\dfrac{1}{x}\) notice that this is just \(y=5\cdot \dfrac{1}{x-1}\) which is a horizontal shift, and a vertical stretch, of \(y=\dfrac{1}{x}\)

:)

Answer 8

Be careful on #2. It's not in lowest form. :)

Answer 9

Reduce to figure out what the graph LOOKS like, but you still have to leave out of the domain, any x-values that make the ORIGINAL (not reduced) expression undefined.

Answer 10

That's only half of the graph, you know? (Just making sure you know you aren't finished yet).

And you need to get the graph over closer to the VA. Right now it looks like you have x=0 (the y-axis) as a vertical asymptote.