Video Transcript
Which of the following processes would you use to obtain the graph of 𝑦 equals 𝑓 of negative three 𝑥 from the graph of 𝑦 equals 𝑓 of 𝑥? (a) Reflect the graph in the 𝑥-axis. (b) Reflect the graph in the 𝑦-axis. (c) Stretch the graph in the horizontal direction by a factor of three. Or (d) stretch the graph in the horizontal direction by a factor of one-third.
Let’s begin by considering the algebraic transformation represented by the four processes given. We should recall that when a transformation has a horizontal effect, it corresponds to making a change to the variable, whereas when a transformation has a vertical effect, it corresponds to making a change to the function itself. Reflecting a graph in the 𝑥-axis has a vertical effect, and it corresponds to the transformation 𝑔 of 𝑥 is mapped to negative 𝑔 of 𝑥. On the other hand, reflecting a graph in the 𝑦-axis has a horizontal effect, and it corresponds to the transformation 𝑥 is mapped to negative 𝑥. So wherever we had 𝑥 before, we replace it with negative 𝑥.
Stretching a graph in the horizontal direction by a factor of three corresponds to the variable 𝑥 being replaced with one-third 𝑥, whereas stretching the graph in the horizontal direction by a factor of one-third corresponds to the variable 𝑥 being replaced with three 𝑥. We want to see then how we obtain 𝑓 of negative three 𝑥 from 𝑓 of 𝑥. Well, we can see that what has changed is the variable. We had 𝑓 of 𝑥 and now we have 𝑓 of negative three 𝑥. 𝑥 has been negated and multiplied by three.
Looking at the transformations we wrote down, we can see that this corresponds to a reflection in the 𝑦-axis and a stretch of the graph in the horizontal direction by a factor of one-third. The order in which these two particular transformations are applied is unimportant. The same result is achieved either way around. So, we found that to obtain the graph of 𝑦 equals 𝑓 of negative three 𝑥 from the graph of 𝑦 equals 𝑓 of 𝑥, we need to reflect the graph in the 𝑦-axis and stretch the graph in the horizontal direction by a factor of one-third, which corresponds to performing processes (b) and (d).
