Answer: g(x) = (-2/3)x^2
=====================================================
Step-by-step explanation:
The blue parent function has a positive coefficient of 1. The purple g(x) function is a reflection of f(x) over the x axis, so everything is now negative. The coefficient must be negative as well.
But the answer is simply not g(x) = -x^2 because plugging x = 3 does not lead to y = -6 as the point (3,-6) shows.
Let's say the coefficient is k for now. So we have y = kx^2
Plug in x = 3 and y = -6. Solve for k
y = kx^2
-6 = k(3)^2
-6 = k*9
9k = -6
k = -6/9
k = -2/3
So we update y = kx^2 into y = (-2/3)x^2
Meaning that g(x) = (-2/3)x^2 is the equation of the purple curve.
Plug x = 3 into g(x) to find that
g(x) = (-2/3)x^2
g(3) = (-2/3)(3)^2
g(3) = (-2/3)(9)
g(3) = -6
which is the output we want, so this confirms we have the correct coefficient.