### Video Transcript

Which equation matches the
graph? Option (A) π¦ equals π₯ minus two
cubed minus one. Option (B) π¦ equals π₯ plus two
cubed minus one. Option (C) π¦ equals π₯ plus two
cubed plus one. Or option (D) π¦ equals π₯ minus
two cubed plus one.

We might begin by noticing that
this function looks very similar to the standard cubic function π of π₯ is equal to
π₯ cubed, sometimes known as π¦ equals π₯ cubed. We can sketch π¦ equals π₯ cubed
alongside the given function. The graph of π¦ equals π₯ cubed has
an inflection point at zero, zero. The inflection point of the given
function is at negative two, negative one. We could therefore say that the
function π¦ equals π₯ cubed must have been translated two units left and one unit
down. Both of these functions have the
same steepness, and they have not been reflected, so there are no further
transformations.

We can recall that a cubic function
in the form π¦ is equal to π times π₯ minus β cubed plus π is a transformation of
π¦ equals π₯ cubed for π, β, and π in the real numbers and π not equal to
zero. In this form, the value of π
indicates the dilation scale factor and a reflection if π is less than zero;
thereβs a horizontal translation of β units right and a vertical translation of π
units up. We perform these transformations
with the vertical stretch first, horizontal translation second, and vertical
translation third.

In this question, the graph has not
been reflected or dilated, so π is equal to one. Next, we identified that this graph
has a translation of two units left. Because this cubic function form
has a horizontal translation of β units to the right, then this means that our value
of β must be negative. And so β is equal to negative
two. Finally, we identified that there
must be a vertical translation of one unit down. Since this form gives us a vertical
translation in terms of units upwards, then our value of π must be negative. So itβs negative one.

Now all we need to do is fill in
the values of π, β, and π into this cubic function form. When we simplify, we get the
equation π¦ equals π₯ plus two cubed minus one. This is the equation given in
answer option (B).