See if we can solve this problem. In this problem, we have a function h(x), which is a transformation of the original function f(x) = x3. We're told in this transformation, the function is reflected over the x-axis and then shifted down two units. We are asked to write an equation for h(x) and sketch a graph of the function h(x). So let's begin.
First off, we should recognize what type of transformations are happening here. In this first case, I see that we have a reflection over the x-axis. Recall from previous videos that when we reflect over the x-axis, our function f(x) becomes -f(x). This is something we learned in the video on reflections. We can also see here we have a shift down 2 units. Since we're only shifted down, this means we're only being vertically shifted. When this happens, our original function f(x) becomes f(x) + k, where k represents the vertical shift. So these are the two situations we need to watch out for in this problem.
We’ll start off by examining the function here, which is f(x) = x3. The first transformation that happens is a reflection over the x-axis, and when this happens, the function becomes negative. So x3 is going to become -x3. This is the first transformation where we reflect over the x-axis. The next transformation is the shift down 2 units. Since we're shifting down 2 units, this k represents the vertical shift. Since we're shifted down, that means k is going to be -2 for the downward shift. Adding k to this equation, our transformation becomes -x3 - 2. This right here is our transformed function h(x), and this is what the new equation is going to look like. This is how you can determine the equation for h(x), and that's part a of this problem.
We are also asked to sketch a graph of h(x) as well. To do this, let's once again look at the transformations we have. This is the original function right here. First off, we have a reflection over the x-axis. Recall that when reflecting over the x-axis, you can imagine folding your graph over the x-axis, as if increasing it like a piece of paper. So this portion of the function is going to go down like this, and then this portion of the function is going to fold up like this. So this is what the function will look like when we reflect over the x-axis, but we're not quite done yet because notice that we've also been shifted down 2 units. So the graph is not going to be centered at the origin, it’s actually going to be centered down, 2 units right here at -2. So our final graph is going to look like this for the transformed function.
This is our transformation h(x) as a graph, and this is the corresponding equation. This is how you can deal with multiple transformations on a single function. Hope you found this helpful.