Video: Sketching Curves from their Roots

Which of the following is the graph of 𝑓(π‘₯) = (π‘₯ + 2)(π‘₯ βˆ’ 2)Β²? [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 π‘₯ equals π‘₯ plus two times π‘₯ minus two squared?

So 𝑓 of π‘₯ is a cubic polynomial, and it’s a nicely factored form for us. We can use these factors to work out where the π‘₯-intercept of the graph should be. The factor of π‘₯ plus two tells us that there should be an π‘₯-intercept at negative two and the other factor of π‘₯ minus two squared tells us that there is another π‘₯-intercept at π‘₯ equals two. And furthermore, these are the only two π‘₯-intercepts.

We can therefore eliminate option A, which has another π‘₯-intercept at π‘₯ equals zero or thereabouts. We are therefore left with the options B, C, D, and E which have π‘₯-intercepts at negative two and two, and they don’t have any extra π‘₯-intercepts anywhere else. To decide between these four remaining options, we’re going to have to look at the type of root at two and negative two.

We have a single factor of π‘₯ plus two; this is a simple root therefore at π‘₯ equals negative two. And so the graph should cut the π‘₯-axis at π‘₯ equals negative two; it should cross from one side to the other. We’re not yet sure which way it should cross the π‘₯-axis β€” from below to above or from above to below β€” but it must cross.

And as a result, we can eliminate options B and D, where the graph does not cross the π‘₯-axis at π‘₯ equals negative two; the graph there only touches the π‘₯-axis. The factor of π‘₯ minus two, on the other hand, is repeated as an π‘₯ minus two squared. And so near π‘₯ equals two, the graph of the function must touch, but not cross the π‘₯-axis either from above or below; we’re not sure yet. Unfortunately, this does not help us choose between our two remaining options C and E. Both of these options show the graph touching, but not crossing the π‘₯-axis at π‘₯ equals two.

To decide between these two options, we can expand the brackets in the definition of 𝑓 of π‘₯ to get π‘₯ plus two times π‘₯ squared minus four π‘₯ plus four, which is π‘₯ cubed minus two π‘₯ squared minus four π‘₯ plus eight. The coefficient of π‘₯ cubed is positive; it’s one, and so we expect a cubic curve in this orientation and not in the opposite orientation because when π‘₯ is very large, 𝑓 of π‘₯ will behave very much like π‘₯ cubed, which grows as π‘₯ grows. And so the graph of 𝑓 of π‘₯ will point up and to the right and not down and to the left.

We can also see that the 𝑦-intercept is plus eight, which is positive of course. So the graph of 𝑓 of π‘₯ should cross the 𝑦-axis above the π‘₯-axis. Either way we can see that option E cannot be our answer, and we’re left only with option C. And we can see that this option has the orientation of the curve correct and also that the 𝑦-intercept of this curve is positive as required.

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