Which of the following graphs
represents the equation 𝑓 of 𝑥 equals negative 𝑥 minus one squared?
And there are five different graphs
to choose from.
So, we’re given a function
equation, and we need to identify the correct graph for this equation. There are actually two ways we can
answer this. We could draw a table of values and
plot the relevant graph from that. Alternatively, we can use what we
know about the graph of the most basic quadratic equation alongside knowledge of
function transformations to identify the correct graph. Let’s use this method.
Consider the function equation 𝑔
of 𝑥 equals 𝑥 squared. The graph of this equation is a
symmetric parabola with a turning point at zero, zero and a vertical line of
symmetry given by the 𝑦-axis. We can see that the equation we
have, 𝑓 of 𝑥 equals negative 𝑥 minus one squared, looks a little like this. In fact, this is equivalent to
taking the original equation, 𝑔 of 𝑥 equals 𝑥 squared, subtracting one from the
value of 𝑥, and then multiplying the entire expression by negative one. Hence, we can say that 𝑓 of 𝑥
equals negative 𝑔 of 𝑥 minus one.
So, how does this help? Well, we know that given a general
function 𝑦 equals 𝑓 of 𝑥, we can map that onto 𝑦 equals 𝑓 of 𝑥 plus 𝑎 by a
translation negative 𝑎, zero, in other words, a translation 𝑎 units left. Similarly, given a general function
𝑦 equals 𝑓 of 𝑥, we can map that onto 𝑦 equals negative 𝑓 of 𝑥 by reflecting
in the 𝑥-axis. This means we map 𝑔 of 𝑥 onto 𝑓
of 𝑥 by translating one unit right and reflecting in the 𝑥-axis.
Notice that the order in which we
perform these actions here doesn’t matter; we’ll achieve the same result either
way. If this was not the case, we would
need to perform a horizontal translation before a reflection of any kind. That gives us this graph. We can see that that corresponds to
option (E) here. The graph that represents the
equation 𝑓 of 𝑥 equals negative 𝑥 minus one squared is (E).