### Video Transcript

Which of the following processes would you use to obtain the graph of π¦ equals π of negative three π₯ from the graph of π¦ equals π of π₯? (a) Reflect the graph in the π₯-axis. (b) Reflect the graph in the π¦-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.

Letβs 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 π₯-axis has a vertical effect, and it corresponds to the transformation π of π₯ is mapped to negative π of π₯. On the other hand, reflecting a graph in the π¦-axis has a horizontal effect, and it corresponds to the transformation π₯ is mapped to negative π₯. So wherever we had π₯ before, we replace it with negative π₯.

Stretching a graph in the horizontal direction by a factor of three corresponds to the variable π₯ being replaced with one-third π₯, whereas stretching the graph in the horizontal direction by a factor of one-third corresponds to the variable π₯ being replaced with three π₯. We want to see then how we obtain π of negative three π₯ from π of π₯. Well, we can see that what has changed is the variable. We had π of π₯ and now we have π of negative three π₯. π₯ 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 π¦-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 π¦ equals π of negative three π₯ from the graph of π¦ equals π of π₯, we need to reflect the graph in the π¦-axis and stretch the graph in the horizontal direction by a factor of one-third, which corresponds to performing processes (b) and (d).