Question Video: Transforming Graphs with Arithmetic Operations Mathematics

Consider the graph of the linear function π(π₯). Which of the following is the graph of the function π(2π₯)? [A] Graph A [B] Graph B [C] Graph C [D] Graph D

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Video Transcript

Consider the following graph of the linear function π of π₯. Which of the following is the graph of the function π evaluated at two π₯? Option (A), (B), (C), or (D).

In this question, weβre given the graph of a linear function π of π₯. We need to use this to determine the graph of the function π evaluated at two π₯ from a list of options. And thereβs several different ways we could do this. The easiest way is to recall exactly what transformation π evaluated at two π₯ is from π evaluated at π₯. We recall for a constant π, π evaluated at ππ₯ is a horizontal stretch by a factor of one over π of our function π of π₯. And one way of seeing this would be to try inputting values of π₯ into our function π evaluated at ππ₯.

For example, if we substitute π₯ is equal to π΄ over two and we use our value of π equal to two, then we get π evaluated at two times π΄ over two which is of course just π evaluated at π΄. And in this case, we actually know π evaluated at π΄. π evaluated at π΄ is the π₯-intercept. So this is just equal to zero. Therefore, in our new curve, our π₯-intercept is now going to be at π΄ divided by two. Weβve halved the distance of our π₯-intercepts. In other words, weβve stretched it horizontally by a factor of one over π. Now, we could start eliminating options to answer this question. However, we could also sketch the curve π¦ is equal π evaluated at two π₯.

First, we know it has an π₯-intercept at the value of π΄ divided by two. Next, because this is a horizontal stretch of our original curve, it must pass through the value of π. Because this lies on our vertical axis, when we stretch it horizontally, itβ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, weβ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 π of π₯, we were not only able to determine which of four given graphs is the graph of the function π evaluated at two π₯, we were also able to sketch this ourselves by using the fact that π evaluated at two π₯ will be a horizontal stretch by a factor of one-half. We were able to show that the correct sketch given was option (B).