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# Graphing rational functions 3

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

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• You made it perfectly clear. What about oblique (I hope I used good term, english is not my native language) asymptotes?
• What made the video example have a horizontal asymptote was the fact that the numerator (x) & denominator (x^2 -16) polynomials both had the same degree (their greatest exponents; in the example video, this was degree 2). More generally, the numerator & denominator have different degrees. Here are the possibilities:

Case 1: Numerator degree = Denominator degree. Effect: Horizontal Asymptote, as in the video. The horizontal asymptote line is at the y-value that equals the ratio of the numerator & denominator coefficients (multiplying numbers) of their highest power terms; example: y = (12x^5 + 7x^2 - 8x + 9)/(4x^5 - 13x^4 + 55) has the horizontal asymptote being the line y = 3 because 12/4=3.

Case 2: Numerator degree < Denominator degree. Effect: Horizontal Asymptote is the x-axis. Reason: For huge |x| values, the bigger denominator power creates such a huger value than what the numerator makes, so when we divide the numerator by this great value, the value of the division is very close to 0. Example: (3x^2 - 5)/(x^4 +7x - 8) produces, approximately, when x=1000, the division 3,000,000/1,000,000,000,000 = 0.000003, which is indeed quite close to 0. If You don't like approximating (but You should!-- it's a powerful technique), the precise computation is 2,999,995/1,000,000,006,992.

Case 3: Numerator degree > Denominator degree. This case includes your question (oblique asymptote [yes, "oblique" is the correct term, meaning 'slanted']), and it also includes other more complicated cases.

METHOD: Use long-division of polynomials to divide the numerator by the denominator, obtaining a Quotient polynomial Q(x) and a Remainder polynomial R(x). When You do this, the Remainder polynomial always ends up having a degree that is less than the degree of the original denominator polynomial. In elementary school, when we divide, say, 17/3 we obtain a quotient of 5 with a remainder of 2; all this information is contained in the equations 17 = 5*3 + 2 or (equivalently) 17/5 = 3 + 2/3. In this numerical example, notice that the Remainder 2 turns into the Remainder FRACTION 2/3 when we write the second equivalent equation. Similar things happen with long-division of polynomials: Starting with y = n(x)/d(x) where n(x) is whatever the numerator polynomial is, and where d(x) is whatever the denominator polynomial is, when we do long-division we obtain the two equivalent results n(x) = d(x)*Q(x)+R(x) and n(x)/d(x) = Q(x)+R(x)/d(x). This is just like the 17/5=3+2/3 example above!

Let's get concrete with the graph of an example of this nature. Starting with y = (4x^2+8x-60)/(x-6), the long-division [see the video that's a few steps after the one here, for how to do long-division] yields the results Q(x)=4x+32 and R(x)=132. Therefore the original equation is equivalent to

y = (4x+32) + 132/(x-6). Now we are ready to learn about what's happening with the graph: Observe that when x is huge in magnitude (i.e., getting closer to either + or - infinity), then the Remainder Fraction 132/(x-6) gets ever closer to 0 in value. Example, when x=-1,000,000 we have 132/-1,000,006 which is pretty tiny. Because of this, we conclude that when x is huge (either + or -), then our graph is really close to the graph of y = (4x+32)+0. The straight line y=4x+32 is what is called an OBLIGUE ASYMPTOTE for the rational function y = (4x^2+8x-60)/(x-6). Recapping what this means: The graph of y = (4x^2+8x-60)/(x-6) gets really close to the slant line y = (4x+32) as we move further & further to the right (x approaching + infinity) or further & further to the left (x approaching - infinity).

GENERAL RESULT: After conducting long-division we convert any rational function y = n(x)/d(x) into the equivalent form y = Q(x) + R(x)/d(x). As x becomes huge (either + or -), the value of the Remainder Fraction R(x)/d(x) always approaches the value 0, because the process of long-division always yields a Remainder whose degree is less than the degree of the denominator d(x). THEREFORE, the graph of the quotient, y = Q(x), always gives an asymptote for the original rational function. This asymptote is properly called the Main Asymptote or Quotient Asymptote. Every rational function has a Main Asymptote. [It's possible for a rational function to have NO vertical asymptotes. Example y = 2x^3/(x^2+1).]

Extra #1: In the example above we obtained the Remainder polynomial to be R(x)=132, a constant function that didn't depend upon the value of x. If we had started out with the degree of the denominator d(x) to be 2 or more, then it can happen that the Remainder polynomial R(x) is a function that has x in it. Without doing a complete example, if d(x) was degree 2, then it would be possible to obtain R(x)=2x-44, which is equivalent to R(x)=2(x-22). Notice that for the special x-value of x=22, we have R(22)=0, so for this special value the general fact that n(x)/d(x) = Q(x) + R(x)/d(x) would become n(x)/d(x) = Q(x) + 0 WHEN x=22. This means that the graph we want for y = n(x)/d(x) and the graph of the Main Asymptote y = Q(x) have the very same y-value WHEN x=22, so our desired graph of y = n(x)/d(x) actually intersects the (dotted) graph of the Main Asymptote y=Q(x). The example here had R(x)=2(x-22), causing our function to cross through the graph of the Main Asymptote when x=22. IF instead we had had a situation where d(x) was degree 3, and where it so happened that R(x)=3(x-50)^2, then (because of the square on the (x-50) here) our graph of y = n(x)/d(x) would just KISS the graph of the Main Asymptote y=Q(x)WHEN x=50; this is just like how a parabola, say y = 4(x-7)^2, just kisses the x-axis when x=7. The key issue when deciding if a root, Z, of R(x) [i.e., an x-value for which R(Z)=0] is a Crossing Point -or- a Kissing Point of intersection between the actual graph and the graph of the Main (Quotient) Asymptote is whether the root value, Z, had an Even or Odd power on (x - Z) in the factored form of R(x). Above we had an example with R(x)=2(x-22)^1 and the other example with R(x)=3(x-50)^2. The term for the exponent on a root's factor is the MULTIPLICITY of the root. So if R(x) has an ODD multiplicity root, the graph of the desired function n(x)/d(x) CROSSES the graph of the Main Asymptote y=Q(x) at that particular root. If R(x) has an EVEN multiplicity root, the graph of the desired function n(x)/d(x) KISSES the graph of the Main Asymptote y=Q(x) at that particular root. Note that this distinction between crossing behaviour & kissing behaviour is just like the issue for roots of simple polynomial graphs.

Extra #2: Recall the general idea that y = n(x)/d(x) = Q(x) + R(x)/d(x), and that because long-division yields that R(x) always has lower degree than the degree of d(x), the value of the Remainder Fraction R(x)/d(x) always approaches 0 when the value of x approaches + or - infinity. The nice consequence of this is that our desired graph approaches the graph of the Main Asymptote y = Q(x)when the value of x approaches + or - infinity. We'd like to know a little more!: we'd like to know if the true graph of y = n(x)/d(x) = Q(x) + R(x)/d(x) ends up being ABOVE or BELOW the graph of the asymptote y = Q(x). This question is easy to answer-- we just need to determine if the Remainder Fraction R(x)/d(x) [which we know shrinks to 0 as x gets huge] is positive or negative for huge values of x. This is quite easy! Suppose, for example, that we have the Remainder Fraction R(x)/d(x)= (-3x^2+5x+4)/(x^3+4x^2-7x+8). When x is huge in magnitude, the highest power terms in the numerator & denominator dominate the value of the fraction: R(x)/d(x) is approximately (-3x^2)/(x^3) which reduces to -3/x. As x grows to be huge positive this fraction shrinks to 0 as expected, but we see now that it will be a tiny NEGATIVE value. THEREFORE, we'll have n(x)/d(x)=Q(x)+(tiny negative) which causes our desired graph of n(x)/d(x) to be a bit less than the Quotient asymptote; our graph approaches the Quotient Asymptote from below it, WHEN x is approaching (+)infinity. On the other end of the picture, when x approaches (-)infinity, note that the approximated Remainder Fraction, -3/x, is now a tiny POSITIVE value. THEREFORE, we'll have n(x)/d(x)=Q(x)+(tiny positive) which causes our desired graph of n(x)/d(x) to be a bit greater than the Quotient asymptote; our graph approaches the Quotient Asymptote from above it, WHEN x is approaching (+)infinity.

LOOKING BACK: The analysis of a rational function via long-division to consider
y = n(x)/d(x) = Q(x) + R(x)/d(x) actually applies perfectly to all 3 Cases above, including the simplest cases of horizontal asymptotes. For the above Case 1 where the numerator n(x) and denominator d(x) both have the same degree (say this degree is N), we saw that there was a horizontal asymptote at the value y=A/B where n(x)=Ax^N + other junk, and where d(x)=Bx^N + other junk. Observe that the long-division yields this same conclusion, because Q(x)=A/B. For the above Case 2 with the numerator n(x) having a lesser degree (say L) than the degree (say M) of the denominator d(x), causing the x-axis to be the horizontal asymptote, we have the following long-division analysis. The long-division of n(x)/d(x) yields a Quotient Q(x)=0 because we're trying to divide a bigger degree polynomial into a lesser degree polynomial. Once again, we have our Main Asymptote, now given by the equation y = 0 (which is indeed the x-axis). So, overall, it's not really remembering 3 different cases; all 3 cases can be thought of through the long-division mentality.
• 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.
• 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?
• If you think about it, the function x^2/(x^2-16) can never equal 1, because you know (x^2)/(x^2) = 1 (since anything over itself equals 1). So if the denominator is (x^2)-16, the denominator will always be a little smaller than the numerator, therefore the graph will never actually reach y=1.
I'm not quite sure about what your second question means. But in any case, X can equal any number except 4 and -4 (for this particular function). Sal's graph shows this as well.
Hope this helps! :)
• 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.