Video Transcript
If 𝑓 of 𝑥 is equal to 𝑥 cubed is changed to 𝑓 of 𝑥 is equal to 𝑥 minus one cubed plus five, how is the graph transformed?
In this question, we’re transforming the standard cubic curve 𝑦 is equal to 𝑥 cubed into the curve 𝑦 is equal to 𝑥 minus one cubed plus five. We want to determine the exact series of transformations which transform this curve into the other. And there’s several different ways we can do this. For example, we can do this in several different ways by using the given function.
One way of doing this is to note that we’re subtracting a value of ℎ from the input values of 𝑥 and then we’re adding a value of 𝑘 to the output of the function. We can then recall subtracting a value of ℎ from the input values of a function will translate it ℎ units in the horizontal direction. And adding 𝑘 to the outputs of the function will translate the curve 𝑘 units in the vertical direction. This is only one way of answering the question. However, we can also do this by using a more standard method.
We can notice that the transformed function is given in the form 𝑎 multiplied by 𝑥 minus ℎ all cubed plus 𝑘. And we can recall this is the standard form for transformations of the cubic curve. The sign of 𝑎 tells us whether the curve is reflected in the 𝑥-axis and the magnitude of 𝑎 tells us the vertical dilation of the curve. And as before, the magnitudes of ℎ and 𝑘 tell us the amount that the curve is transformed horizontally and vertically. And the signs of these two constants tell us the directions.
In our case, the value of 𝑎 is one, the value of ℎ is one, and the value of 𝑘 is five. Since the value of 𝑎 is one, we can note that there is a vertical stretch of factor one. In other words, there’s no vertical stretch. Next, the value of ℎ is one. This is positive, so this is a translation one unit to the right. Finally, the value of 𝑘 is five. This is positive, so this is a translation five units upwards.
This then gives us our final answer. If the function 𝑓 of 𝑥 is equal to 𝑥 cubed is changed to the function 𝑓 of 𝑥 is equal to 𝑥 minus one cubed plus five, then the cubic curve 𝑦 is equal to 𝑥 cubed is horizontally shifted right by one unit and vertically shifted up by five units.
