Estimating an oblique asymptote graphically
About this video
The video checks from a graph whether y = -1/3 x + 14/3 is an oblique (slanted) asymptote of H(x). It plots the line using its y-intercept 14/3 (4 and 2/3) and slope -1/3, and sees that it matches the line the graph approaches, while noting that this is an estimate, not a proof.
You'll be able to
- describe an oblique asymptote as a slanted line that a graph approaches.
- sketch a line from its y-intercept and slope to compare it with a graph.
- explain why checking from a graph is an estimate rather than a proof.
Key words
- Oblique asymptote
- A slanted (not horizontal or vertical) line that a graph gets closer to as x becomes very large or very negative.
- Slope
- How much y changes for each step in x; -1/3 means down 1 for every 3 to the right.
- Y-intercept
- Where a line crosses the y-axis, the value of y when x = 0.
Transcript
- [Instructor] We're asked to determine if Y is equal to negative one third X plus 14 third is an oblique asymptote of the rational function H of X. So pause this video, have a go at it before we do it together. All right, now, an oblique asymptote is an asymptote that is at an angle, is one way to think about it.
It's slanted, so if I just eyeball it, it looks like there is an oblique asymptote that looks something like that. And let's see if that matches up to Y is equal to negative one third X plus 14 over three. We're gonna have to do some close inspection here. So one thing we can do is we can say, all right, this has a Y intercept of 14 over three.
Let me do this in a slightly different color. Of 14 over three. 14 over three is the same thing as four and two thirds. So we should have a Y intercept of four and two thirds, which is right about there, and then we have a slope of negative one third. So every time we move three, we should go down one.
Every time we go three, we should go down one. Every time we go three, we should move down one. And of course, we could do that in the other direction. And this is looking pretty good. I could go in the other direction. If we go three to the left, then we should go up one to five and two thirds.
Three to the left, we should go up one. Three to the left, we should go up one. And so if I were to connect these dots, I don't have a straight edge here, but this is my hand-drawn version, it really does look like this Y is equal to negative one third X plus 14 thirds is the oblique asymptote.
At least I can do that from, it looks that way. I'm not proving it here, but it looks like that.
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