Graphs of rational functions: y-intercept | Mathematics III | High School Math | Khan Academy

Let F of x = A * x^n + Bx + 12 over C * x^m + Dx + 12, where M and N are integers and A, B, C, and D are unknown constants.

All right, this is interesting! Which of the following is a possible graph of y equal F of x? They tell us that dashed lines indicate asymptotes, and they give us four choices here. I encourage you, like always, to pause this video, give it a go, and see if you can figure out which of these graphs could be the graph for y equal F of x, where F of x is given this information about F of x.

Now, let's work through this together. This is really interesting; they didn't give us a lot. They didn't even tell us what the exponents on x are, and they haven't even given us the coefficients. All they've told us is this 12 here. This 12 looks like a pretty big clue, so what can that tell us? The way they've written it, we really can't deduce any zeros for the function. We really can't deduce what x values make the numerator equal zero or what x values make the denominator equal zero.

So, it's going to be hard for us to deduce what the zeros of the function are or what the removable discontinuities are or what the vertical asymptotes are. Just this 12 sitting here does tell us one thing: what happens when x equals zero? Because when x equals 0, every other term in this rational expression is just going to be equal to zero. So, we can figure out F of 0.

F of 0 is going to be equal to A * 0^n, well, that's just going to be 0, plus B * 0, well, that's just going to be 0, plus 12 over C * 0^m power, well, that's just going to be 0, plus D * 0, which is going to be zero, and then we have our 12 there. So, we're actually able to figure out what F of 0 is: it's 12 over 12, or 1.

So, we actually know the y-intercept for this function. Let’s see if that’s enough information for us to figure out if any of these choices could be the graph of y equal F of x.

So, let's see here. For choice A, our y-intercept is at 2. When x equals 0, our graph goes through 2, so we can rule that out. The y-intercept needs to be 1.

Now, let’s see choice B. It does have a y-intercept, it looks like just eyeballing it at 1 when x equals 0; y is 1, so this looks interesting. Choice C has its y-intercept at y equal to 1, so once again, we can rule that out. Choice D has no y-intercept at all.

So, that was enough information, and lucky for us, because they really didn’t give us a lot more information than just being able to evaluate F of 0. We weren’t able to figure out any of the other zeros or the vertical asymptotes or the removable discontinuities. So, we definitely feel good about that choice.