Consider the graph of the function 𝑦 is equal to 𝑥 plus two all cubed minus two. Write down the coordinates of the point of symmetry of the graph, if it exists.
In this question, we’re given the graph of a function, and we can see that our function is a cubic polynomial. We’re asked to just write down the point of symmetry of the graph, if it exists. And the most important thing to note is that we’re asked for a point of symmetry. For example, we’re not being asked for a line of symmetry. And cubic polynomials never have lines of symmetry because the end behavior of their curves are always opposite, and reflection through a line of symmetry will switch the end behaviors.
So instead, we’re looking for a point of symmetry. This will be rotational symmetry since we rotate about a point. And there’s a few different ways we could go about doing this. The easiest way is to note that the function we’re given is actually a translation of the cubic polynomial. Since we add two to our 𝑥 input values, we translate the curve two units to the left and then we subtract two from the outputs. This translates the graph two units down. So this is just the standard cubic curve 𝑦 is equal to 𝑥 cubed translated two units left and two units down. We can see this in the diagram.
We can then recall the curve 𝑦 is equal to 𝑥 cubed has rotational symmetry about the origin, the point zero, zero. And it’s worth pointing out the reason for this is the function 𝑓 of 𝑥 is equal to 𝑥 cubed is an odd function, which means 𝑓 evaluated at negative 𝑥 is equal to negative 𝑓 of 𝑥. And we can then use this to show there will be rotational symmetry. In particular, if the point with coordinates 𝑥, 𝑦 lies on this curve, then the point with coordinates negative 𝑥, negative 𝑦 will also lie on the curve.
We can then recall translation does not affect the shape of the curve. So we’ve just translated the point of symmetry. We’ve translated it two units to the left and two units down. That’s from the origin to the point with coordinates negative two, negative two. Therefore, we were able to show the graph of the function 𝑦 is equal to 𝑥 plus two all cubed minus two has a point of rotational symmetry at the point negative two, negative two.