Which of the following is a graph of 𝑓 of 𝑥 equals 𝑥 plus two times 𝑥 minus two squared?
So 𝑓 of 𝑥 is a cubic polynomial, and it’s a nicely factored form for us. We can use these factors to work out where the 𝑥-intercept of the graph should be. The factor of 𝑥 plus two tells us that there should be an 𝑥-intercept at negative two and the other factor of 𝑥 minus two squared tells us that there is another 𝑥-intercept at 𝑥 equals two. And furthermore, these are the only two 𝑥-intercepts.
We can therefore eliminate option A, which has another 𝑥-intercept at 𝑥 equals zero or thereabouts. We are therefore left with the options B, C, D, and E which have 𝑥-intercepts at negative two and two, and they don’t have any extra 𝑥-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 𝑥 plus two; this is a simple root therefore at 𝑥 equals negative two. And so the graph should cut the 𝑥-axis at 𝑥 equals negative two; it should cross from one side to the other. We’re not yet sure which way it should cross the 𝑥-axis — from below to above or from above to below — but it must cross.
And as a result, we can eliminate options B and D, where the graph does not cross the 𝑥-axis at 𝑥 equals negative two; the graph there only touches the 𝑥-axis. The factor of 𝑥 minus two, on the other hand, is repeated as an 𝑥 minus two squared. And so near 𝑥 equals two, the graph of the function must touch, but not cross the 𝑥-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 𝑥-axis at 𝑥 equals two.
To decide between these two options, we can expand the brackets in the definition of 𝑓 of 𝑥 to get 𝑥 plus two times 𝑥 squared minus four 𝑥 plus four, which is 𝑥 cubed minus two 𝑥 squared minus four 𝑥 plus eight. The coefficient of 𝑥 cubed is positive; it’s one, and so we expect a cubic curve in this orientation and not in the opposite orientation because when 𝑥 is very large, 𝑓 of 𝑥 will behave very much like 𝑥 cubed, which grows as 𝑥 grows. And so the graph of 𝑓 of 𝑥 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 𝑥 should cross the 𝑦-axis above the 𝑥-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.