Video: Sketching Curves from their Roots

Which of the following is the graph of 𝑓(π‘₯) = (π‘₯ βˆ’ 2)Β³? [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 𝑓 π‘₯ equals π‘₯ minus two all cubed?

To start off solving this problem, what I’ve actually done is I’ve actually drawn a sketch of the graph 𝑦 equals π‘₯ cubed or 𝑓π‘₯ equals π‘₯ cubed. And as you can see, this actually goes through the origin. To consider how 𝑓π‘₯ of π‘₯ minus two all cubed might differ from the 𝑓π‘₯ equals π‘₯ cubed graph, we’re actually gonna look at some translations and some rules for these.

The first of which is that 𝑓π‘₯, in parentheses, plus π‘Ž, that will give us a shift of π‘Ž units in the 𝑦-direction. What does this mean in practice? Well, it actually means that all of our 𝑦-coordinates would actually add π‘Ž to each of those to actually give us what our shift would be.

Okay. We’re gonna have a look at another translation. And our second translation rule tells us that 𝑓 of π‘₯ plus π‘Ž, this time the plus π‘Ž is in the parentheses. And this means a shift of negative π‘Ž units in the π‘₯-direction. Remember you need to pay particular attention to the fact that it’s negative π‘Ž units. And that what it means in practice is that actually we’re gonna subtract π‘Ž from each of the π‘₯-coordinates in order to actually show how our graph has shifted.

Okay. So if we look back at our function, so we’ve got 𝑓π‘₯ equal to π‘₯ minus two all cubed. Using our translation rule, therefore we can say that it’s gonna be a shift of negative negative two units in the π‘₯-direction. And actually, looking at that because it says it’s negative negative two units. What that actually means is we’re actually gonna have a shift of plus two units in the π‘₯-direction. And what this actually means in practice, is that therefore we’re gonna add two to all of our π‘₯-coordinates.

Okay, great! Let’s go back to our original graph and see what this would do. So as you can see in the graph that’s shown in pink, it’ll actually shift the whole graph two units to the right because we’ve actually added two to every π‘₯-coordinate. And therefore, fantastic! We can actually see which is the correct answer. We look back at π‘Ž. So π‘Ž is our correct answer because we can actually see that the intercept there would be at two.

So π‘Ž is the correct graph of 𝑓π‘₯ is equal to π‘₯ minus two all cubed.

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