Video Transcript
The graph of the function 𝑓 of 𝑥 equals root 𝑥 is first reflected symmetrically over the 𝑦-axis, then shifted up two units and right by three, and finally horizontally stretched by two to obtain the graph of the function 𝑔 of 𝑥. Write an equation for 𝑔 of 𝑥.
Let’s begin by considering the sequence of four geometric transformations that are applied to the function 𝑓 of 𝑥. First, 𝑓 of 𝑥 is reflected over the 𝑦-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 𝑔 of 𝑥 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. Let’s consider a reflection over the 𝑦-axis first. This transformation has a horizontal effect. And it corresponds to the algebraic transformation 𝑥 is mapped to negative 𝑥. In other words, wherever we have 𝑥 in the function, we’re going to replace it with negative 𝑥.
So, we started with the function 𝑓 of 𝑥 equals the square root of 𝑥. And to reflect over the 𝑦-axis, we replace 𝑥 with negative 𝑥, giving the function the square root of negative 𝑥. Next, we recall that a vertical shift or translation by 𝑘 units corresponds to mapping the function ℎ of 𝑥 to ℎ of 𝑥 plus 𝑘. So, to shift the graph up by two units, we need to add two to the entire function. We’ve already replaced 𝑥 with negative 𝑥, so we’re adding two to the square root of negative 𝑥. And we find that after the first two transformations have been applied, the equation of this function is the square root of negative 𝑥 plus two.
Next, let’s consider the horizontal shift. A horizontal shift or translation 𝑘 units to the right corresponds to changing the variable from 𝑥 to 𝑥 minus 𝑘. So, to shift the function right by three units, we need to replace 𝑥 with 𝑥 minus three. We need to be a little careful here. In the previous stage, we had the square root of negative 𝑥. So when we replace 𝑥 with 𝑥 minus three, we need to make sure we multiply the entirety of the expression 𝑥 minus three by negative one. After the first three transformations then, the equation we have is the square root of negative 𝑥 minus three plus two. Then of course we can distribute the parentheses to give the square root of negative 𝑥 plus three plus two.
The final transformation is another horizontal transformation, so once again it affects the 𝑥-variable. We recall that to perform a horizontal stretch by a scale factor of 𝑘, the variable 𝑥 must be replaced with one over 𝑘 multiplied by 𝑥. So, to perform a horizontal stretch with a scale factor of two, we replace 𝑥 with a half 𝑥. This gives the square root of negative a half 𝑥 plus three plus two.
We’ve now performed all four transformations in the specified order. And we found that the equation of the function 𝑔 of 𝑥 is 𝑔 of 𝑥 is equal to the square root of negative a half 𝑥 plus three plus two.
