Describe the transformation necessary to transform the graph of f(x) into that of g(x)

Question

f(x)=x^2

g(x)=1/3x^2

Answer 1

assuming you meant g(x) = (1/3)x^2

then f(x) is compressed vertically, which is the same as expanded horizontally, The parabola is made flatter, by a factor of 3, or compressed vertically by a factor of 1/3.

if you compressed further to 1/3 to 1/30 to 0, the parabola becomes a flat straight horizontal line = the x axis, y=0

but the way you have written it, g(x)=1/3x^2 is literally = 1/(3x^2) which is not a parabola. graph it, and it's two separate hyperbola branches with the axes as asymptotes, x=0 as a vertical asymptote, y=0 as a horizontal asymptote, for both branches. the parabola vertex becomes the intersection of the asymptote lines, the two symmetric halves of the parabola become two symmetric halves of the hyperbola. g(x) = 1/(3x^2) inverts f(x)=x^2 into x^-2, then compresses it vertically by 1/3

Answer 2

The function f(x) will need to be dilated by a factor of 1/3 in order to transform it into g(x). The graph of g(x) will have the same vertex and will appear be a compressed (less than) version of f(x).