Question Video: Finding the Equation of a Cubic Function Following a Dilation and a Reflection Mathematics

The function 𝑓(π‘₯) = (π‘₯ + 2)Β³ + 3 is stretched in the vertical direction by a scale factor of 2 then reflected in the 𝑦-axis. Write the equation of the transformed function 𝑔(π‘₯).

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Video Transcript

The function 𝑓 of π‘₯ equals π‘₯ plus two cubed plus three is stretched in the vertical direction by a scale factor of two then reflected in the 𝑦-axis. Write the equation of the transformed function 𝑔 of π‘₯.

There are two transformations that have been applied to our function 𝑓 of π‘₯. Firstly, it’s stretched in the vertical direction; then it’s reflected. Now, we might recall that in order to stretch a function in the vertical direction, we need to multiply that entire function by the scale factor. So 𝑓 of π‘₯ will be mapped onto two times 𝑓 of π‘₯ to achieve this. And given some function 𝑓 of π‘₯, the corresponding function 𝑓 of negative π‘₯ represents a reflection in the 𝑦-axis of that original function. Now, the order in which we apply these is important. So we will follow the order in the question, starting with the vertical stretch.

We have 𝑓 of π‘₯ equals π‘₯ plus two cubed plus three, and we’re going to multiply the entire function by two. When we do, we find that two 𝑓 of π‘₯ is equal to two times π‘₯ plus two cubed plus three. Then we can distribute this two across the parentheses and we have that two times 𝑓 of π‘₯ is two times π‘₯ plus two cubed plus six. Now that we’ve performed the stretch in the vertical direction, we’re going to perform a reflection in the 𝑦-axis.

To achieve this, we change the π‘₯ to a negative. In other words, we essentially multiply the value of π‘₯ by negative one. Since we’re starting off with our already transformed function, we need to change two 𝑓 of π‘₯ to two 𝑓 of negative π‘₯. And all we do here is we multiply the value of π‘₯ by negative one. So two 𝑓 of negative π‘₯ is two times negative π‘₯ plus two cubed plus six. And so we have the equation of the transformed function, which we can now write as 𝑔 of π‘₯. 𝑔 of π‘₯ is two times negative π‘₯ plus two cubed plus six.

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