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# Graphing rational functions 3

Sal graphs y=(x^2)/(x^2-16). Created by Sal Khan and CK-12 Foundation.

## Want to join the conversation?

• what is an ASYMPTOTE??
• The asymptote is a 'line' on the coordinate plane where the graph of a rational equation approaches but never actually intersects. There are 3 types of asymptotes: vertical, horizontal, and oblique.
• Why was 4.01 picked when findind y? Could 7 be picked or another number?
• Probably because he wanted to make it much more noticeable than going down by 1. Instead, he went down by one decimal place to make it very obvious that is works. Yes, Sal could have used 7, or any number in fact; he just wanted to state his point.
(1 vote)
• can asymptotes of a hyperbola be considered as tangents?
• Also, by definition, tangents touch a shape (hyperbola, circle, etc.) in exactly one spot. A function will never touch an asymptote. Now, towards the extremes of the function, will an asymptote appear to be tangent to an equation? Sure. But it's not, and won't behave the same way.
• how would you write rational functions from a graph or sata like

a zero at 2
vertical asymptote at x=-1
horizontal asymptote at y=1
???
• Would (3x)^2 be 3x^2 or 9x^2 because you have to distribute it?
• the parentheses around the 3x part is very important. (3x)^2 is different than 3x^2. The first one means you are squaring the whole thing, whereas the second means you are just squaring x and then multiplying by 3.
(3x)^2 = (3x)(3x) = 9x^2
• Can someone explain why if the numerator > denominator there is no horizontal asymptotes?
• This has to do with the nature of horizontal asymptotes. They tell you about the end-behavior of functions (i.e. the limit as x-> infinity)

When the degree of the numerator is larger than the degree of the denominator, that means that the value of the numerator is going to increase much more quickly than the value of the demoninator.

This, in turn, means that the value of the function will increase without bounds as x approachs infinity. So there is no number that the ends will get closer and closer to and thus no horizontal asymptote.
• This question has been lingering in my head ever since I learned the concept of asymptoptes, but it is quite a vague/unclear one.

What happens when the line reaches a point where if it attempts to move 1 unit closer to the asymptote, it will equal the asymptote?

Let's say that we have the function y=f(x) and it is a downward curve approaching the horizontal asymptote y=3 from the above. When the function reaches 2.999(gazillions of 9's)99, it is so close to y=3 that if the value of f(x) decreases by 0.000(gazillions of 0's)01 (moving 1 unit to the right on the x-axis), it will equal 3.
I know that at such a point, it cannot equal 3, so what happens there?
• It is not an actual requirement that the function does not ever equal the asymptote at any value as x approaches infinity. That does tend to be the case, but it is not a requirement.

An asymptote is what the function approximates when x (or y, depending on what kind of asymptote) approaches infinity.

So, there is no rule saying that that the function and the asymptote cannot meet for some values of x. Once x gets large, it is usually the case that the function and the asymptote won't ever be exactly the same, but there do exist functions in which that is the case (but only for some values of x, never all values of x).

However, the case that you described won't actually happen. Even when you have 2.999..... on and on for countless trillions of 9's, you still won't have it =3. You never run out of "one more 9" that you can add.

Though, of course, for all practical purposes, once the function gets so close to the asymptote that there is no real-world, useful distinction between the two, we can just say they have become essentially equal.
• Are there diagonal (for lack of a better term) asymptotes (i.e., non-horizontal or vertical)?
• Yes, there is. It's called oblique or Slant asymptotes.

• Why does Y never equal 1? That is, why is the horizontal Asymptote a 1?
Why can X never equal something greater than -4 but less than 4--that is, something in between the two vertical Asymptotes?