Find g(x) where g(x) is the reflection across the x-axis of f(x)=x^2

Answer 1

To reflect a graph, f(x) over the x-axis, you take -f(x).

So if f(x)=x^2, then -f(x) is -x^2.

Then g(x)=-x^2 is the reflection of your function f(x) over the x-axis.

Answer 2

The function f(x)=x² is a parabola facing upward with its vertex at (0,0), and including the points (1,1), (2,4), (-1,1), and (-2,4). A function that is a reflection across the x-axis compared to f(x) would be the mirror image of f(x). It would have the opposite y-values for each x-value. It would also have its vertex at the same place as f(x). Therefore, it would be a parabola pointing downward, with the points (1, -1), (2,-4), (-1,-1), and (-2,-4). To get the opposite values for y, you simply need to place a negative in front of the x². So the function would be g(x) = - x².

Answer 3

Your question does not specify whether you have to do this algebraically or just through logic or visualization, so let's go about it the easy way:

Start by graphing y = x². This is the parent parabola function with vertex at (0,0). OK so far?

Reflection across the x-axis basically means that any given x value will generate the opposite y value from that in the original function. So, in the original, you have (1,1). In the reflection, you have (1,-1). This transformation will be true for all points, except that the vertex (0,0) will remain the same, since there's no negative version of zero.

Plot a few more points, and you'll see the mirror image of the original. What you've already learned about transformations tells you that the original shape is retained (no stretch or compression), and the vertex has had no up-down or left-right movement. The only transformation is the "flip," which means your transformation is the negative version of the original, y = -x².

Best wishes!