Question Video: Sketching Curves from their Roots Mathematics • 10th Grade

Which of the following is the graph of ๐‘“(๐‘ฅ) = (๐‘ฅ โˆ’ 2)ยณ? [A] Graph A [B] Graph B [C] Graph C [D] Graph D [E] Graph E.

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Video Transcript

Which of the following is the graph of ๐‘“ ๐‘ฅ equals ๐‘ฅ minus two all cubed?

To start off solving this problem, what Iโ€™ve actually done is Iโ€™ve actually drawn a sketch of the graph ๐‘ฆ equals ๐‘ฅ cubed or ๐‘“๐‘ฅ equals ๐‘ฅ cubed. And as you can see, this actually goes through the origin. To consider how ๐‘“๐‘ฅ of ๐‘ฅ minus two all cubed might differ from the ๐‘“๐‘ฅ equals ๐‘ฅ cubed graph, weโ€™re actually gonna look at some translations and some rules for these.

The first of which is that ๐‘“๐‘ฅ, in parentheses, plus ๐‘Ž, that will give us a shift of ๐‘Ž units in the ๐‘ฆ-direction. What does this mean in practice? Well, it actually means that all of our ๐‘ฆ-coordinates would actually add ๐‘Ž to each of those to actually give us what our shift would be.

Okay. Weโ€™re gonna have a look at another translation. And our second translation rule tells us that ๐‘“ of ๐‘ฅ plus ๐‘Ž, this time the plus ๐‘Ž is in the parentheses. And this means a shift of negative ๐‘Ž units in the ๐‘ฅ-direction. Remember you need to pay particular attention to the fact that itโ€™s negative ๐‘Ž units. And that what it means in practice is that actually weโ€™re gonna subtract ๐‘Ž from each of the ๐‘ฅ-coordinates in order to actually show how our graph has shifted.

Okay. So if we look back at our function, so weโ€™ve got ๐‘“๐‘ฅ equal to ๐‘ฅ minus two all cubed. Using our translation rule, therefore we can say that itโ€™s gonna be a shift of negative negative two units in the ๐‘ฅ-direction. And actually, looking at that because it says itโ€™s negative negative two units. What that actually means is weโ€™re actually gonna have a shift of plus two units in the ๐‘ฅ-direction. And what this actually means in practice, is that therefore weโ€™re gonna add two to all of our ๐‘ฅ-coordinates.

Okay, great! Letโ€™s go back to our original graph and see what this would do. So as you can see in the graph thatโ€™s shown in pink, itโ€™ll actually shift the whole graph two units to the right because weโ€™ve actually added two to every ๐‘ฅ-coordinate. And therefore, fantastic! We can actually see which is the correct answer. We look back at ๐‘Ž. So ๐‘Ž is our correct answer because we can actually see that the intercept there would be at two.

So ๐‘Ž is the correct graph of ๐‘“๐‘ฅ is equal to ๐‘ฅ minus two all cubed.

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