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Video: Sketching Curves from their Roots

Chris O’Reilly

Which of the following is the graph of 𝑓(𝑥) = −(𝑥 +1)³(𝑥 − 2)? [A] Graph A [B] Graph B [C] Graph C [D] Graph D [E] Graph E.


Video Transcript

Which of the following is the graph of 𝑓𝑥 is equal to negative 𝑥 plus one all cubed multiplied by 𝑥 minus two?

Well, the first thing we can do is actually eliminate some graphs. And we can do that when we look, first of all, at the 𝑥 to the power of four term. From the first parentheses, we’re actually gonna have a negative 𝑥 cubed term. And we get that because, within the parenthesis itself, we have an 𝑥 term. And the parentheses are actually cubed. So that’s gonna give us 𝑥 cubed.

And then we’ve actually got a negative before the parentheses. So it’s gonna give us negative 𝑥 cubed. And this is gonna be multiplied by an 𝑥 term from our second parentheses. And that’s gonna give us a negative 𝑥 to the power of four term. And that’s actually the highest-order 𝑥 term that we have. So if we use this to consider the shape of our graph, then actually a positive 𝑥 power four term kind of give us this upright 𝑤-type shape. And a negative 𝑥 power four would give us an 𝑚-type shape. Oh, we see it depends on the function itself. But that gives you a general shape.

So now if we have a look, we’ve got a negative 𝑥 power four. So that means we’re actually looking for a graph that would be in a general form of an 𝑚 shape. What that means, because of this, we’ll be able to rule out any graphs that haven’t got that shape. So we’ve got the sort of positive 𝑥 power four shape.

So looking at our graphs, we can actually rule out graph C because it actually has that positive 𝑥 power four shape. Okay, we’ve got four left. Now we need to decide which one of these is going to be the graph that we’re actually looking for for our function. So now we’re gonna actually look at the roots of our function.

Well, if we take a look, we can actually see that if we’re to make our first parentheses equal to zero, then 𝑥 will be equal to negative one. So we’ve actually got a root of negative one. And because this parentheses is actually cubed, it means we’re actually gonna have a repeated root here, which is gonna be useful when we’re actually looking to find the graph.

In our second parentheses, we actually have 𝑥 minus two. So if we’re making that equal to zero, 𝑥 will be equal to two. So therefore, that is another one of our roots of our function. So therefore, we’re now able to rule out two more graphs because they don’t have the roots negative one for 𝑥 and two for 𝑥. So we can rule out 𝐵 because that had negative two and one as its roots and 𝐷 because its roots were negative one and one.

So now we need to actually decide between graph A and graph E. Well, actually, if we look at graph A, we can actually see that there’s actually a repeated root. You can see that because of the shape of the graph at this point. And actually we want a repeated root because it says that it’s 𝑥 plus one all cubed. So therefore, it’s looking very likely that graph A is the correct graph.

What we’re gonna do now is have a look at graph E because there’s actually something here that confirms that. We can see that graph E does in fact have a root at negative one. And it also has a root at two. However, there is also a root because it touches the 𝑥-axis at one. So therefore, we can rule out graph E. And we can in fact say that graph A must be the correct graph because it has a root at negative one that’s repeated and it also has a root at two. And therefore, we can say the graph of 𝑓𝑥 equals negative 𝑥 plus one all cubed multiplied by 𝑥 minus two is graph A.