### Video Transcript

The function π 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 π¦-axis. Write the equation of the
transformed function π of π₯.

There are two transformations that
have been applied to our function π of π₯. Firstly, itβs stretched in the
vertical direction; then itβs 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 π of π₯ will be mapped onto two
times π of π₯ to achieve this. And given some function π of π₯,
the corresponding function π of negative π₯ represents a reflection in the π¦-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 π of π₯ equals π₯ plus two
cubed plus three, and weβre going to multiply the entire function by two. When we do, we find that two π 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 π of π₯ is two times π₯ plus two
cubed plus six. Now that weβve performed the
stretch in the vertical direction, weβre going to perform a reflection in the
π¦-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 weβre starting off with our
already transformed function, we need to change two π of π₯ to two π of negative
π₯. And all we do here is we multiply
the value of π₯ by negative one. So two π 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 π of π₯. π of π₯ is two times negative π₯
plus two cubed plus six.