Video Transcript
Consider the function 𝑓 of 𝑥 is
equal to 𝑥 minus one all squared plus two, where 𝑥 is greater than or equal to one
and less than ∞. There are three parts to this
question. Which of the graphs represents the
function 𝑓 of 𝑥? Which of the graphs represents the
reflection of 𝑓 of 𝑥 on the 𝑥-axis? And which of the graphs represents
the reflection of 𝑓 of 𝑥 on the 𝑦-axis?
On the figure given, there are five
graphs labeled (A) to (E). We firstly need to identify which
graph represents the function 𝑥 minus one all squared plus two where 𝑥 lies on the
left-closed, right-open interval from one to ∞. We can immediately exclude options
(B) and (D) as these are defined on the interval from negative ∞ to negative
one. To work out which of the other
three graphs represents the function, we can investigate the behavior of the
function at a particular value. It is sensible to begin with the
boundary values. And in this case, we will calculate
𝑓 of one. When 𝑥 is equal to one, 𝑓 of 𝑥
is equal to one minus one all squared plus two. This is equal to two, which means
that the graph of the function must pass through the point one, two. This is only true of graph C. The graph which represents the
function 𝑓 of 𝑥 is (C).
The second part of our question
asks us which of the graph represents the reflection of 𝑓 of 𝑥 in the 𝑥-axis. Reflecting the point one, two in
the 𝑥-axis gives us the point one, negative two. It is therefore clear that graph A
represents the reflection of 𝑓 of 𝑥 on the 𝑥-axis. Reflecting the point one, two in
the 𝑦-axis gives us the point negative one, two. This confirms that graph B is a
reflection of 𝑓 of 𝑥 on the 𝑦-axis. Our three answers are (C), (A), and
(B).
