If I wanted to change the y-intercept of a quadratic from (0,2) to (0,5), which transformation explains what happened to the graph?

Many transformations will affect the y-intercept of a quadratic. However, the most obvious one is translation, which moves the whole quadratic up (or down if k<0) by an amount k.

The transformation is

T (0, k) f (x) - > f (x) + k

For example, the quadratic f (x) = x^2+2x+2 (y-intercept = 2) will be transformed to f (x) - >f (x) + (5-2) = f (x) + 3=x^2+2x+5.