 In this video, we'll provide the solution to question number three for practice exam number one for math 1210, in which case we're asked to recognize which function formula gives us the picture here on the right, for which we'll probably recognize this one very quickly. This is the graph of course, f of x equals x cubed. So that would be answer D. Now, if you're not sure, I mean, these are, I should say, these are just graphs that you wanna memorize. So like if you go through them one by one, f of x equals one, this is a constant function. It'll look like a horizontal line. f of x equals x, this is the identity function. This will just be the diagonal line that goes through the origin. If we talk about the parabola, f of x equals x squared, well, like I said, this is a parabola. It would look something like this, right? This would be y equals x squared. Now, if we choose any even monomial, it'll have the same basic shape, but as the power gets bigger, bigger, bigger, like if you're looking at y equals x to the fourth, it's gonna get flatter near the origin and steeper when you get away from the origin. So this would be like something like y equals x to the fourth. Continuing in this vein, if we were talking about an odd monomial, like what f of x equals x cubed, well, the even monomials are gonna be even functions. That is, they're symmetric with respect to the y-axis. The odd functions, like f of x equals x cubed, this will be symmetric with respect to the origin. If you were to take a half spin, you get the exact same graph again. So this would be the graph of y equals x cubed. If you wanna do like x to the fifth, it's gonna get steeper near the origin, excuse me, flatter near the origin, and then steeper away from the origin. So you get something like this, y equals x to the fifth, although that's not an option on our list here. If we were considering the square root of x, the square root of x, it's a half parabola, which will concave to the right. And so this would be the square root of x. If you want a higher degree, then that is if you take like the fourth root, it's gonna get steeper near the origin and then flatter as you get away from the origin. Right, it's sort of a subtle point there. If we took the cube root of x, well, this would look, well, basically it's just gonna be the reflection of this graph across the diagonal line of y equals x there, in which case you get a picture that looks something like the following. This would be f of x equals x cubed. You wanna do like the fifth root or the seventh root. Again, you make it steeper near the origin, flatter away from the origin. That's the basic shape there. f of x equals the absolute value of x. That's our classic v shape. To the right of the y-axis, it looks just like y equals x. To the left of the y-axis, it looks like y equals negative x. So you get that classic v shape. And then f of x equals one over x. You're gonna get this reciprocal function. It has a vertical asymptote at x equals zero. It has a horizontal asymptote at y equals zero. You get this basic shape. As you increase the power to like three or five or seven, you'll see that these lines will get much more, well, it'll get closer to its horizontal asymptote. And it'll get a little bit farther away from the vertical asymptote in terms of the graph steepness and things like that. And the only other one I would mention here is if you're looking at f of x equals one over x squared, the graph would look something like this, y equals one over x squared. It still has a horizontal asymptote at the x-axis and a vertical asymptote at the y-axis. But you'll notice that both functions approach positive infinity as it approaches the vertical asymptote. So they're both touching infinity on the top. And then of course, if you switch this to be like the fourth power or the sixth power, again, you'll see that it'll converge towards its horizontal asymptote faster. It'll converge towards its vertical asymptote just a little bit slower, I suppose.