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
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).