Christian is rewriting an expression of the form y = ax2 + bx + c in the form y = a(x – h)2 + k. which of the following must be true?
h and k cannot both equal zero
k and c have the same value
the value of a remains the same
h is equal to one half –b

"h and k cannot both equal zero" -- yes, it can. if the vertex of a parabola is at (0, 0), there's nothing incorrect/invalid about that!!

"k and c have the same value" -- k and c do not have the same value. "k" is the y-value of the vertex and c is the constant in your quadratic equation, and the constant is not necessarily the y-value.

"the value of a remains the same" -- this is true. the a's in your equations are the same values, because the a-value is the coefficient of the x-variable in both equations. y = a(x - h)^2 and y = ax^2 -- both of these have a applying to your x-variables.

"h is equal to one half -b" -- this isn't true. the formula for calculating the x value of the vertex (h is the x-value of the vertex) is h = (-b/2a). -b/2a is not the same as one half -b because this answer choice doesn't involve the a-value.