Video Transcript
If 𝑓 of 𝑥 is equal to 𝑥 cubed is changed to 𝑦 is equal to 0.1 multiplied by 𝑥 minus 10 all cubed, how is the graph transformed?
In this question, we’re transforming the standard cubic function 𝑓 of 𝑥 equals 𝑥 cubed into the given form 0.1 multiplied by 𝑥 minus 10 cubed. We need to determine how the graph of this curve is transformed. And the easiest way to do this is to note the function, which is the image of the transformation, is given in the form 𝑎 multiplied by 𝑥 minus ℎ cubed plus 𝑘. In this case, our value of 𝑎 is 0.1, the value of ℎ is 10, and the value of 𝑘 is zero. And this is a useful form to help us determine the transformations because the values of 𝑎, ℎ, and 𝑘 tell us how the curve is transformed.
First, we can recall the value of 𝑎 tells us the vertical dilation of the graph. In particular, this is a stretch in the 𝑦-direction by a factor of the absolute value of 𝑎. And if 𝑎 is negative, we also need to reflect the graph in the horizontal axis. In this case, our value of 𝑎 is positive and equal to 0.1. So we’re going to need to stretch the graph of 𝑦 is equal to 𝑥 cubed in the 𝑦-direction by a factor of 0.1.
Next, the value of ℎ tells us the horizontal translation of our graph. In particular, if the value of ℎ is positive, we translate the graph ℎ units to the right. And if the value of ℎ is negative, then we translate the graph the absolute value of ℎ units to the left. In this case, the value of ℎ is 10. So we’re going to need to translate the graph 10 units to the right.
Finally, our value of 𝑘 is zero. So we have no vertical translation of the graph, which then gives us our final answer. If the curve 𝑦 is equal to 𝑥 cubed is changed to 𝑦 is equal to 0.1 times 𝑥 minus 10 cubed, then the curve is stretched by a factor of 0.1 in the 𝑦-direction and shifted right by 10 units.
