Video: Transforming Graphs with Arithmetic Operations

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

In this question, we’re given the graph of a linear function 𝑓 of π‘₯. We need to use this to determine the graph of the function 𝑓 evaluated at two π‘₯ 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 π‘₯ is from 𝑓 evaluated at π‘₯. We recall for a constant 𝑏, 𝑓 evaluated at 𝑏π‘₯ is a horizontal stretch by a factor of one over 𝑏 of our function 𝑓 of π‘₯. And one way of seeing this would be to try inputting values of π‘₯ into our function 𝑓 evaluated at 𝑏π‘₯.

For example, if we substitute π‘₯ 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 π‘₯-intercept. So this is just equal to zero. Therefore, in our new curve, our π‘₯-intercept is now going to be at 𝐴 divided by two. We’ve halved the distance of our π‘₯-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 π‘₯.

First, we know it has an π‘₯-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 π‘₯, we were not only able to determine which of four given graphs is the graph of the function 𝑓 evaluated at two π‘₯, we were also able to sketch this ourselves by using the fact that 𝑓 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).

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