### Video Transcript

Which of the following is a graph of π of π₯ equals π₯ plus two times π₯ minus two squared?

So π of π₯ is a cubic polynomial, and itβs 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 donβt have any extra π₯-intercepts anywhere else. To decide between these four remaining options, weβre 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. Weβre 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; weβre 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 π 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; itβs one, and so we expect a cubic curve in this orientation and not in the opposite orientation because when π₯ is very large, π of π₯ will behave very much like π₯ cubed, which grows as π₯ grows. And so the graph of π of π₯ will point up and to the right and not down and to the left.

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