Consider the following graph of the
linear function 𝑓 of 𝑥. Which of the following is the graph
of the function 𝑓 evaluated at two 𝑥? Option (A), (B), (C), or (D).
In this question, we’re given the
graph of a linear function 𝑓 of 𝑥. We need to use this to determine
the graph of the function 𝑓 evaluated at two 𝑥 from a list of options. And there’s several different ways
we could do this. The easiest way is to recall
exactly what transformation 𝑓 evaluated at two 𝑥 is from 𝑓 evaluated at 𝑥. We recall for a constant 𝑏, 𝑓
evaluated at 𝑏𝑥 is a horizontal stretch by a factor of one over 𝑏 of our function
𝑓 of 𝑥. And one way of seeing this would be
to try inputting values of 𝑥 into our function 𝑓 evaluated at 𝑏𝑥.
For example, if we substitute 𝑥 is
equal to 𝐴 over two and we use our value of 𝑏 equal to two, then we get 𝑓
evaluated at two times 𝐴 over two which is of course just 𝑓 evaluated at 𝐴. And in this case, we actually know
𝑓 evaluated at 𝐴. 𝑓 evaluated at 𝐴 is the
𝑥-intercept. So this is just equal to zero. Therefore, in our new curve, our
𝑥-intercept is now going to be at 𝐴 divided by two. We’ve halved the distance of our
𝑥-intercepts. In other words, we’ve stretched it
horizontally by a factor of one over 𝑏. Now, we could start eliminating
options to answer this question. However, we could also sketch the
curve 𝑦 is equal 𝑓 evaluated at two 𝑥.
First, we know it has an
𝑥-intercept at the value of 𝐴 divided by two. Next, because this is a horizontal
stretch of our original curve, it must pass through the value of 𝑏. Because this lies on our vertical
axis, when we stretch it horizontally, it’s not going to move. Finally, remember, our original
function is a linear function. When we stretch this by a factor of
one-half in the horizontal direction, we’re also going to get a linear function. So we can connect these two points
with a straight line, giving us the following sketch. And of our four options, we can see
this is given by option (B).
Therefore, given the graph of the
linear function 𝑓 of 𝑥, we were not only able to determine which of four given
graphs is the graph of the function 𝑓 evaluated at two 𝑥, we were also able to
sketch this ourselves by using the fact that 𝑓 evaluated at two 𝑥 will be a
horizontal stretch by a factor of one-half. We were able to show that the
correct sketch given was option (B).