Consider the function \(g(x) =-5x^2 + 100x- 450\). First you need to simplify it by expressing the perfect square:
\(g(x) =-5x^2 + 100x- 450=-5(x^2-20x+90)=-5(x^2-2\cdot x\cdot 10+10^2-10^2+90)=-5((x-10)^2-100+90)=-5(x-10)^2+50.\)
So you get \(g(x) =-5(x-10)^2+50\).
Now you can consider all transformations that have been applied to the graph of \(f(x) = x^2\) to produce the graph of g(x):
- translate the graph of the function \(f(x)\) right 10 units - this transformation gives you the graph of the function \(f_1(x)=(x-10)^2\);
- reflect graph of the function \(f_1(x)\) about the x-axis. This transformation gives you the graph of the function \(f_2(x)=-(x-10)^2\);
- stretch the graph of the function \(f_2(x)\) in the y-direction by multiplying the whole function by 5, then you get the function \(f_3(x)=-5(x-10)^2\);
- translate the graph of the function \(f_3(x)\) up 50 units, then you get the graph of the function \(g(x) =-5(x-10)^2+50\).
From the algorithm above you can conclude that choices B, E and F are correct.
