### Video Transcript

The graph of the function π of π₯ equals root π₯ is first reflected symmetrically over the π¦-axis, then shifted up two units and right by three, and finally horizontally stretched by two to obtain the graph of the function π of π₯. Write an equation for π of π₯.

Letβs begin by considering the sequence of four geometric transformations that are applied to the function π of π₯. First, π of π₯ is reflected over the π¦-axis. Then, the graph is shifted up by two units and shifted right by three units. Finally, the graph is horizontally stretched by a scale factor of two. And we obtain the graph of the function π of π₯ whose equation we wish to find. To do so, we need to consider what effect each of these transformations has on the equation of a function when we apply them in the given order.

We recall first that when a transformation has a horizontal effect, it corresponds to making a change to the variable, whereas when a transformation has a vertical effect, it corresponds to making a change to the function itself. Letβs consider a reflection over the π¦-axis first. This transformation has a horizontal effect. And it corresponds to the algebraic transformation π₯ is mapped to negative π₯. In other words, wherever we have π₯ in the function, weβre going to replace it with negative π₯.

So, we started with the function π of π₯ equals the square root of π₯. And to reflect over the π¦-axis, we replace π₯ with negative π₯, giving the function the square root of negative π₯. Next, we recall that a vertical shift or translation by π units corresponds to mapping the function β of π₯ to β of π₯ plus π. So, to shift the graph up by two units, we need to add two to the entire function. Weβve already replaced π₯ with negative π₯, so weβre adding two to the square root of negative π₯. And we find that after the first two transformations have been applied, the equation of this function is the square root of negative π₯ plus two.

Next, letβs consider the horizontal shift. A horizontal shift or translation π units to the right corresponds to changing the variable from π₯ to π₯ minus π. So, to shift the function right by three units, we need to replace π₯ with π₯ minus three. We need to be a little careful here. In the previous stage, we had the square root of negative π₯. So when we replace π₯ with π₯ minus three, we need to make sure we multiply the entirety of the expression π₯ minus three by negative one. After the first three transformations then, the equation we have is the square root of negative π₯ minus three plus two. Then of course we can distribute the parentheses to give the square root of negative π₯ plus three plus two.

The final transformation is another horizontal transformation, so once again it affects the π₯-variable. We recall that to perform a horizontal stretch by a scale factor of π, the variable π₯ must be replaced with one over π multiplied by π₯. So, to perform a horizontal stretch with a scale factor of two, we replace π₯ with a half π₯. This gives the square root of negative a half π₯ plus three plus two.

Weβve now performed all four transformations in the specified order. And we found that the equation of the function π of π₯ is π of π₯ is equal to the square root of negative a half π₯ plus three plus two.