 In this video, we present the solution to question number 14 for practice exam number 3 from math 1050. We're given a polynomial function f of x, which fortunately for us is factored. f of x is negative x plus 7 times 6x plus 1 times x minus 8 squared. We're supposed to indicate all of the x-intercepts and y-intercepts of presence, so let's list them. The x-intercepts, of course, are going to equal. You look at the roots here, so you have a negative 7, you have a negative 1 sixth, and you have a positive 8. It does also mention that we should list the multiplicities of these things. The x plus 7 shows up once, so its multiplicity is 1. The 6x plus 1 shows up once, so its multiplicity is 1. The x minus 8 shows up twice, so its multiplicity is there too. And be aware how this affects the graph. Odd multiplicities, they will cross the x-axis, but even multiplicities will touch it. So as I start preparing my graph here, let's see. 1, 2, 3, 4, 5, 6, 7, 8. So I can draw a picture of a point right here. This is x equals 8, right? Then we have negative 7. 1, 2, 3, 4, 5, 6, 7. So we can do that one right here. And then with negative 1 sixth, this should be really close to the origin. So that's perhaps not perfect, but we're good there. And again, we should label these things. So we're going to get 8 comma 0. We're going to get negative 7 comma 0. And in this point right here, this is going to be negative 1 sixth comma 0. Let's label those on the picture. We could do the labels later, but that's okay. I just put them in now. We need to do the y-intercept. The y-intercept is when you look at f of 0. So that's when, of course, when you plug in 0 for all of the x's. So you get negative 7 times 1 times negative 8 squared, like so. So of course, when you do this, you have negative 7 times 1 of which is negative 7. Negative 8 squared is 64. And then negative 7 times 64, that's equal to negative 448. Like so. Don't worry so much about how big the y-coordinates are. You'll notice when you look at this picture that there are tick marks on the x-axis, but there are none along the y-axis. When you're drawing the graphs of these functions, I don't really care for you to draw to scale in terms of the y-coordinate. That is, if you can perform any type of vertical stretch to the graph whatsoever, or in this case, a vertical compression, don't care so much. Really, all that matters is that the y-intercept will show up below the x-axis. That's again what really matters here, 0 and negative 448. Okay? The other thing I'm going to ask you to do is indicate the leading term. Now to find the leading term, we have to multiply together all of the powers of x with their multiplicities, with their coefficients, put all of those things together, and we see that the leading term will equal negative x times 6x times x squared, or in other words, negative 6x to the fourth. Since this is negative x to the fourth, it's going to be pointing downwards on the right-hand side and on the left-hand side. That will help us as we draw this picture. I'm going to draw this picture from left to right. Since we start downward, we're going to come up to negative 7. Negative 7, we're cross at negative 7, so we have to get a picture that looks something like this. We cross the x-axis. Then at some point, we have to turn around to come to negative 1-6. We also cross at negative 1-6. We'll go through and grab our y-intercept like we do here. Then we're going to come to 8. At 8, we're going to touch the x-axis, but we won't cross it. Then we come down and we're pointing to the bottom right like we were expecting for, of course, the behavior we had before. This gives us a crude, but this is the type of picture we want to do. A crude, but correct picture for the function f of x here.