1.

f(x) is represented by graph B.

g(x) is represented by graph C.

h(x) is represented by graph A.

A quadratic equation is either in the form ax^{2} + bx + c or f(x) = a(x-h)^{2} +k. This is the vertex form where (h, k) is the vertex, or the highest or the lowest point of the graph

If a \neq 0. If a \lt 0 the graph opens up and if a \lt 0 the graph opens down.

f(x) = x^{2}-2 is in the vertex form. Since a = 1, this shows that the graph opens up and the vertex is at (0,-2). The graph B represents this quadratic equation.

g(x) = 2 - x^{2}, tells that a = -1, so the graph opens downward and the vertex is at (0,2). Graph C represents this graph.

h(x) = x^{2} + 3, tells us that a= 1 so our graph opens upward and the vertex is at (0,3). Graph A represents this graph.

2.

As explained above, the equation g(x) = 2-x^{2} represents graph C. The negative sign before x^{2} or -x^{2} says that the value of a is negative and since a\lt 0, the graph opens downward.