Question 1037991
The graph of y = x^2 is a bowl-shaped graph that opens upward. Ie, the bowl is in a position where you can fill it with water and not have it spill over. 


If we stick a negative in front, then the bowl turns upside down. If there was water in it, it would all come out and onto the ground. You can think of the negative as a bad thing which causes a frown shape of the graph. Whereas the original graph is a happy smile (positive coefficient ---> happy graph)



So that's what your teacher means by "curves down". Some books will say "opens downward" and it means the same thing


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The 1/4 indicates that the graph will be wider than normal. It will be horizontally pulled apart more compared to the parent graph.The reason being is that 1/4 = 0.25 is smaller than 1. Whenever the leading term outside like that is smaller than 1, the graph will be stretched in a horizontal direction.


If the term outside is larger than 1, then that means we have a horizontal compression (which leads to vertical stretching).


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The "x-5" portion shifts the graph 5 units to the right. 


If you had x+5, then it would shift 5 units to the left


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There are no other transformations applied (like vertical shifting). 

Here is what the parent graph looks like (<font color=green>green</font>) 
and what the transformed graph looks like (<font color=blue>blue</font>)


{{{ graph( 500, 500, -10, 10, -10, 10, 0,x^2,(-1/4)*(x-5)^2) }}}

Graph of y = x^2 in <font color=green>green</font>


Graph of y = -1/4(x-5)^2 in <font color=blue>blue</font>