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