Select the equation of this curve. Is it (A) 𝑦 equals 𝑥 cubed minus three? (B) 𝑦 equals negative 𝑥 cubed minus three. (C) 𝑦 equals two 𝑥 cubed plus three. Is it (D) 𝑦 equals negative two 𝑥 cubed minus three? Or (E) 𝑦 equals 𝑥 cubed plus three.
Let’s begin by inspecting the shape of our curve. This is the graph of a cubic function, in other words a function of degree three. Now, in fact, the graph of the most basic cubic function 𝑦 equals 𝑥 cubed looks like this. It passes through the point zero, zero. Now, in fact, this point is a point of inflection. It’s a point where the concavity of the graph changes from being convex down to being convex up.
Now, the orientation of the graph we’ve been given is the same as the orientation of the graph of 𝑦 equals 𝑥 cubed. This must mean it has a positive leading coefficient; it must have a positive coefficient of 𝑥 cubed. If the graph had a negative leading coefficient, it would look like a reflection of this graph in the 𝑥-axis. And that’s great because that means we can disregard options (B) and (D) straightaway.
So, we’re now choosing between options (A), (C), and (E). Inspecting our graphs once again, and we can see that the graph of our function has been translated three units downwards. Now, in general, if we’re mapping the graph of 𝑦 equals 𝑓 of 𝑥 onto the graph of 𝑦 equals 𝑓 of 𝑥 plus 𝑎, we perform a vertical translation. We translate that graph by the vector zero, 𝑎 or 𝑎 units up. This means if we take the graph of either 𝑦 equals 𝑥 cubed or 𝑦 equals two 𝑥 cubed and map those onto 𝑦 equals 𝑥 cubed plus three or 𝑦 equals two 𝑥 cubed plus three, respectively, that would be a vertical translation three units up. But we said that our graph was translated three units down. So, that means we can disregard options (C) and (E) completely.
That leaves us just with the graph of 𝑦 equals 𝑥 cubed minus three. Why don’t we check this graph by looking at a point on the curve? Specifically, we can see that our curve passes through the point two, five. So, when we substitute 𝑥 equals two into the correct function, we should get a 𝑦-value of five out. Let’s check this by substituting 𝑥 equals two into the equation 𝑦 equals 𝑥 cubed minus three. When we do, we get 𝑦 equals two cubed minus three. That of course is equal to five, which tells us that the graph of 𝑦 equals 𝑥 cubed minus three must pass through the point two, five. And so the correct answer is (A). The equation of this curve is 𝑦 equals 𝑥 cubed minus three.