Video Transcript
Which equation matches the red graph? Option (A) π¦ is equal to two π₯ cubed. Option (B) π¦ is equal to four π₯ cubed. Option (C) π¦ is equal to 0.5π₯ cubed. Option (D) π¦ is equal to π₯ cubed. Or is it option (E) π¦ is equal to 0.25π₯ cubed?
In this question, weβre given five curves. And we need to determine which of five given equations matches the red curve. And to do this, letβs start by looking at the red curve. We can see that it has a very similar shape to the curve π¦ is equal to π₯ cubed, a cubic curve. In fact, if we look at the five given options, we can see that all five of the options are some multiple of π₯ cubed. Since weβre multiplying π₯ cubed by a constant, this means weβre stretching it vertically. And itβs worth noticing if any of these constants were negative, this would involve a reflection through the π₯-axis. However, we donβt need to worry about that in this case because all five of the given options have a positive scale factor.
There are a few different ways we could go about finding the equation of the red graph. For example, we might be tempted to sketch the curve π¦ is equal to π₯ cubed on the same coordinate axis and then try to determine the series of transformations which transform this graph onto the red curve. And although this would work, this would be quite difficult. Itβs easier to just determine the coordinates of a point which lie on the red graph and determine which of the five given options also has this point on its curve. For example, we can see from the given diagram that the point with coordinates one, four lies on the red curve. And remember in the graph of a function, the π₯-value tells us the input value to the function and the corresponding π¦-coordinate tells us the output of the function.
Therefore, if we say that the red curve is the curve π¦ is equal to π of π₯, then we know π evaluated at one must be equal to four. This then gives us two different ways of answering this question. First, if we say that the function π of π₯ is our cubic function, then we know π evaluated at one is one cubed, which is equal to one. We then want to determine the vertical stretch which will output four instead of one. And to do this, weβre going to need to multiply both sides of the equation by four. In other words, we need to stretch the graph vertically by a factor of four. This would give us that π of π₯ is four π₯ cubed, which is option (B).
This is not the only way of answering this question, however. We can also substitute π₯ is equal to one into the five given options to determine which one contains the point one, four. Substituting π₯ is equal to one into option (A) gives us π¦ is equal to two times one cubed, which we can evaluate is equal to two. So option (A) passes through the point with coordinates one, two. So itβs not the red graph. Substituting π₯ is equal to one into option (B) gives us π¦ is equal to four times one cubed, which we can evaluate is equal to four. So the graph in option (B) passes through the point one, four.
For due diligence, we should also check the other three options, since it might be true that multiple of these curves pass through the point with coordinates one, four. We see when π₯ is equal to one, in option (C), we get π¦ is equal to 0.5. In option (D), when π₯ is equal to one, π¦ is equal to one. And in option (E), when π₯ is equal to one, π¦ is equal to 0.25. Therefore, only the curve given in option (B) π¦ is equal to four π₯ cubed passes through the point with coordinates one, four, which means itβs the only possible equation for the red graph.
