Describe transformations of f(x)=-f(2-x)
suppose we use an example:
e.g.
f(x) = (x-5)^2 + 3 , which is a parabola with vertex (5,3), opening upwards
f(2-x) = (2-x - 5)^2 + 3
f(2-x) = (-x - 3)^2 + 3
f(2-x) = (x+3)^2 + 3 , (just like (-9)^2 = 9^2 )
-f(2-x) = -(x+3)^2 - 3
so we have a reflection in the x-axis, and a horizontal shift of 8 units to the left
http://www.wolframalpha.com/input/?i=plot+y+%3D+(x-5)%5E2+%2B+3
http://www.wolframalpha.com/input/?i=plot+y+%3D+-(x%2B3)%5E2+-+3
best describes the combination of transformations that must be applied to the graph of f(x) = x ^ 2 the graph of g(x) = - x ^ 2 + 1
To understand the transformations of the function f(x) = -f(2-x), we need to break it down into smaller parts.
Let's start with the inner function, g(x) = f(2-x). This function takes a value x, subtracts it from 2, and then applies f to the result. In other words, it reflects the original function f(x) about the vertical line x = 1.
Next, let's look at the outer function, f(x) = -g(x). This function takes the opposite of the value obtained from g(x). We know that g(x) reflects f(x) about x = 1, so by taking the opposite, we reflect it again, this time about the horizontal axis.
Therefore, the combined transformations of f(x) = -f(2-x) are a reflection about the vertical line x = 1, followed by a reflection about the horizontal axis. The resulting graph will have the same shape as the original f(x), but will be reflected both vertically and horizontally.
Note that if you want to visualize these transformations, you can graph the original function f(x), then apply the transformations step by step to see how each transformation affects the graph.