### Video Transcript

The function 𝑓 of 𝑥 equals 𝑥 plus one squared plus two is reflected in the 𝑥-axis and then shifted three units in the positive 𝑦-direction. Write the equation of the transformed function 𝑔 of 𝑥.

There are two transformations being performed to this function: first a reflection in the 𝑥-axis and then a shift three units in the positive 𝑦-direction. That means a shift three units up. We need to recall the effect that each of these transformations have on the equation of a function.

Reflection in the 𝑥-axis, first of all, corresponds to negation of the function. We multiply the entire function by negative one. So, following reflection of the graph of 𝑓 of 𝑥 in the 𝑥-axis, we obtain the function negative 𝑥 plus one squared plus two. Distributing the parentheses gives negative 𝑥 plus one squared minus two.

Next, we recall that a vertical shift or translation three units in the positive 𝑦-direction, so three units up, has the effect of adding the constant three to the entire function. So applying this transformation to the function that has already been reflected in the 𝑥-axis gives negative 𝑥 plus one squared minus two plus three, which simplifies to negative 𝑥 plus one squared plus one. This is the equation of the transformed function 𝑔 of 𝑥.

So we found that if we take the function 𝑓 of 𝑥 equals 𝑥 plus one squared plus two, reflect it in the 𝑥-axis, and then shift it three units in the positive 𝑦-direction, the transformed function we obtain is 𝑔 of 𝑥 equals negative 𝑥 plus one squared plus one.