Question Video: Sketching Curves from their Roots Mathematics

Which of the following is the graph of 𝑓(𝑥) = −(𝑥 + 3)(𝑥 − 1)(𝑥 − 4)? [A] Graph A [B] Graph B [C] Graph C [D] Graph D [E] Graph E


Video Transcript

Which of the following is a graph of 𝑓 of 𝑥 is equal to negative 𝑥 plus three multiplied by 𝑥 minus one multiplied by 𝑥 minus four?

Well, the first thing we’re gonna look at to help us determine which of the graphs it is is actually the negative sign at the front because this is gonna affect the actual shape of the graph. Because what we have is an 𝑥 in each of our factors, so therefore, we’re gonna have 𝑥 multiplied by 𝑥 multiplied by 𝑥, which is gonna give us 𝑥 cubed. So the highest order of 𝑥 is actually gonna be 𝑥 cubed.

Well, if we actually have a positive 𝑥 cubed — so therefore, the coefficient of our 𝑥 cubed term is positive — then the general shape of the graph is like this: so it goes up and down and then up. However, if it’s negative, this is actually inverted, so it goes down then up and then down. And we can see that from the question, our graph is gonna take the form of the second general shape. And that’s because our coefficient of 𝑥 cubed is going to be negative because of the negative in front of our factors.

Okay, so what we can now do is actually rule out any that the shape of the positive 𝑥 cubed. So therefore, what this actually allow us to do is rule out graph B, graph C, and graph D. So we’re just left with graphs A and E. But how we actually can determine which one of these it is? Well, if we take a look at these graphs, there’s one main difference between them. And that’s actually our roots.

We can see that at graph A, the roots are negative three, one, and four. And for graph E, they’re negative four, negative one, and negative three. So now to determine for our function what the roots gonna be, we’re actually gonna have a look at our factors we have inside the parenthesis. So we have 𝑥 plus three, 𝑥 minus one, and 𝑥 minus four.

Now, in order to actually find out what the roots are going to be, what we need to do is set each of these parentheses equal to zero because if one of them is equal to zero, then the answer of each would equal to zero and that’s where it actually crosses the 𝑥-axis.

So we’re gonna start with 𝑥 plus three is equal to zero. So this is gonna give us one of our roots as 𝑥 is equal to negative three. Well, this is actually on both graph A and graph E. So this doesn’t rule any out. Okay, so let’s look at the next parenthesis. So we’re gonna have 𝑥 minus one is equal to zero. Well, this time what we’re gonna do is actually add one to each side of the equation. So therefore, what we’re gonna get is 𝑥 is equal to one. So another root is 𝑥 equal to one.

And if we check back and look at our graphs, we can see that actually well in graph A, we do have a root of one. But in graph E, we don’t cause we actually have negative one. Okay, so on the first, it looks like we’ll definitely gonna have graph A as our graph. What I’m gonna do is I’m gonna check the final factor — so the final root — just to make sure that we actually are correct with graph A.

So having ruled out graph E, we’re gonna do the final factor. So we’ve got 𝑥 minus four is equal to zero. So therefore, we add four to both sides of the equation. And we get 𝑥 is equal to four. So therefore, our final root is four. So if we look up at the graph A, yes, the final root is actually four there as well.

So therefore, we can say that the graph of 𝑓 of 𝑥 is equal to negative 𝑥 plus three multiplied by 𝑥 minus one multiplied by 𝑥 minus four is graph A. And we found that because it took the shape of a negative 𝑥 cubed because we actually had the coefficient of our 𝑥 cubed is going to be negative cause of the negative sign before the factors and also the roots of it were negative three, one, and four, the same as our function.

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