Which equation matches the graph? Is it (A) 𝑦 equals one-half 𝑥 minus one cubed plus four? (B) 𝑦 equals one-third 𝑥 minus one cubed plus four. (C) 𝑦 equals three times 𝑥 minus one cubed plus four. Is it (D) 𝑦 equals two times 𝑥 minus one cubed plus four? Or (E) 𝑦 equals 𝑥 minus one cubed plus four.
Let’s begin by inspecting the graph we’ve been given. We might notice that this is the graph of a cubic function, in other words, a function of degree three. The highest power of 𝑥 is three. Now, in fact, the function 𝑦 equals 𝑥 cubed looks like this. It passes through the origin, the point zero, zero. In fact, the point zero, zero is a point of inflection. It’s the point where the concavity of that graph changes from being convex down to convex up.
Now, if we look at the point of inflection of the graph we’ve been given, we can see that does not lie at the origin. It lies at the point one, four. So this point of inflection has undergone a transformation. Specifically, it has been translated by the vector one, four. That’s one unit right and four units up.
So let’s think about the equation 𝑦 equals 𝑥 cubed. If we want to transform the graph of the equation 𝑦 equals 𝑥 cubed one unit to the right, we subtract one simply from the value of 𝑥. In other words, we get 𝑦 equals 𝑥 minus one all cubed. Then, if we want to translate it four units up, we add four to the entire function. So 𝑦 equals 𝑥 minus one cubed plus four is a translation of the graph 𝑦 equals 𝑥 cubed one right and four up.
Now, unfortunately, this does not eliminate any of our options. We can notice that each option shows this given translation, one unit right and four units up. The difference between each of these functions is the value that sits outside of the parentheses. We have one-half, one-third, three, two, and one. So what does this represent?
Now, in fact, this represents a vertical stretch of the function or a vertical dilation by that value. Normally, this dilation is done before any translation. But since the point of inflection was at zero, zero, it would remain unchanged by a vertical stretch. So, instead, we need to consider some of the other values on our graph. Let’s consider the point with coordinates zero, one. In order to identify the correct vertical stretch, let’s substitute 𝑥 equals zero into each of our equations and check which one yields one for the value of 𝑦.
For equation (A), when 𝑥 is equal to zero, 𝑦 is a half times zero minus one cubed plus four. That’s seven over two, which is not equal to one, so we disregard option (A). Similarly, when 𝑥 is equal to zero, 𝑦 is equal to 11 over three with equation (B). Since that’s not equal to one, we disregard this option. However, for equation (C), when 𝑥 is equal to zero, 𝑦 is three times zero minus one cubed plus four. That’s negative three plus four, which is equal to one as required. And so the equation that matches the graph here is equation (C), 𝑦 equals three times 𝑥 minus one cubed plus four.
For due diligence, let’s double-check option (D) and (E) do not work. In fact, when 𝑥 is equal to zero, option (D) gives 𝑦 equals two and option (E) gives 𝑦 equals three. So we can disregard those options. And that’s confirmed to us that equation (C) matches the graph.