Video Transcript
The graph represents a function 𝑔
of 𝑥 after a vertical positive shift by three units followed by a reflection on the
𝑥-axis. Which of the following represents
the original function 𝑓 of 𝑥? Is it (A) 𝑓 of 𝑥 equals negative
𝑥 plus two cubed minus two? Is it (B) 𝑓 of 𝑥 equals negative
three 𝑥 minus one cubed minus six? Is it (C) 𝑓 of 𝑥 equals 𝑥 minus
one cubed plus two? (D) 𝑓 of 𝑥 equals negative 𝑥
minus four cubed minus two. Or (E) 𝑓 of 𝑥 equals negative
three 𝑥 minus one cubed plus two.
We’re told that 𝑔 of 𝑥 comes from
the original function 𝑓 of 𝑥 after applying two transformations. We can recover the original
function 𝑓 of 𝑥 from 𝑔 of 𝑥 by reversing the transformations but beginning from
the second one. The process will give 𝑓 of 𝑥 as
an expression involving 𝑔 of 𝑥. And then we can finish this problem
by finding an equation for 𝑔 of 𝑥.
So let’s begin with the second
transformation which is a reflection over the 𝑥-axis. And we might remember that given
some function ℎ of 𝑥, the corresponding negative ℎ of 𝑥 is a reflection of the
original graph across the 𝑥-axis. We reverse it by applying the same
transformation. So we take 𝑔 of 𝑥 and we map it
onto negative 𝑔 of 𝑥.
Next, we’ll reverse the first
transformation. That was a vertical positive shift
by three units. The opposite to this is a vertical
negative shift, in other words, shifting or translating the graph three units
down. Now, we achieve that by taking the
function ℎ of 𝑥 and subtracting three from it. Or, in the case of our transformed
function negative 𝑔 of 𝑥, we subtract three from that, so we have negative 𝑔 of
𝑥 minus three. And since we’ve reversed two
transformations, this must be equal to our original function 𝑓 of 𝑥.
So all we need to do now is find
the equation for 𝑔 of 𝑥. We simply look at the shape and we
can already see that it must come from the parent function 𝑦 equals 𝑥 cubed. It’s definitely a cubic graph. But of course if the coefficient of
𝑥 cubed is positive, the cubic graph has the opposite shape. In other words, it looks like this
graph but reflected across the 𝑦-axis.
So to achieve a reflection across
the 𝑦-axis, we make the entire function negative. So 𝑦 equals negative 𝑥 cubed
definitely has the correct shape and orientation. But of course, 𝑦 equals 𝑥 cubed
and 𝑦 equals negative 𝑥 cubed intersect the 𝑦-axis at zero. In fact, this is also the
stationary point of the function or the point of inflection. On our curve, that corresponds to
the point with coordinates one, negative five. So we take 𝑦 equals negative 𝑥
cubed and translate it one unit right and five down. Translating it one unit right
occurs when we subtract one from the value of 𝑥. And to move it five units down, we
simply subtract five from the whole function. So it certainly appears as if we
have the equation for our function 𝑔 of 𝑥.
It might be sensible just to check
this by substituting a couple of the coordinates in. For instance, when 𝑥 is equal to
negative one, 𝑦 is equal to negative negative one minus one cubed minus five, which
equals three as we expected. Similarly, when 𝑥 is equal to
zero, 𝑦 is equal to negative zero minus one cubed minus four, which is negative
four also as we expected. So with some certainty, we can be
assured that we have the correct equation for this graph. So 𝑔 of 𝑥 is negative 𝑥 minus
one cubed minus five, and that means we can now find an expression for 𝑓 of 𝑥.
By replacing 𝑔 of 𝑥 with this
expression in our equation for 𝑓 of 𝑥, we have 𝑓 of 𝑥 is negative negative 𝑥
minus one cubed minus five minus three. And then we distribute the
parentheses to get 𝑥 minus one cubed plus five minus three, which is 𝑥 minus one
cubed plus two. And we can now look at our options
and see that that corresponds to option (C) 𝑓 of 𝑥 is equal to 𝑥 minus one cubed
plus two.
