Question Video: Sketching Curves from their Roots Mathematics • 10th Grade

Video Transcript

Which of the following is the graph of 𝑓 of 𝑥 is equal to negative 𝑥 minus two cubed?

In this question, we’re given a cubic function. So let’s compare this with the standard cubic function 𝑓 of 𝑥 is equal to 𝑥 cubed. We can draw a quick sketch of this function. Let’s recall that a cubic function in the form 𝑓 of 𝑥 is equal to 𝑎 times 𝑥 minus ℎ cubed plus 𝑘 is the transformation of 𝑓 of 𝑥 equals 𝑥 cubed for 𝑎, ℎ, and 𝑘 in the real numbers and 𝑎 not equal to zero.

Here, 𝑎 represents a dilation or reflection, ℎ gives the number of units the graph has been translated in the horizontal direction, and 𝑘 is the number of units the graph is translated in the vertical direction. We perform these transformations with the vertical dilation first, horizontal translation second, and vertical translation third.

So let’s consider the function we were given. Since 𝑓 of 𝑥 is equal to negative 𝑥 minus two cubed, then that means that 𝑎 is equal to negative one. This indicates that there is no dilation, or rather a dilation of scale factor one. However, since 𝑎 is negative, this means there is a reflection of the graph in the 𝑥-axis. If we perform just the reflection, then the graph would look like this in pink.

Next, in the given function, ℎ is equal to two. So this means that there is a translation of two units to the right. This moves the inflection point from zero, zero to two, zero. Therefore, the function would look something like this. It’s always a good idea to mark on any important information onto a sketch. And of course, we know that this graph crosses the 𝑥-axis at two, zero.

We can then give the answer that the graph which shows 𝑓 of 𝑥 is equal to negative 𝑥 minus two cubed is that given in option (E).