Question

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

Answer 1

"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.

Answer 2

Quadratic equations can be expressed in standard form or in vertex form.

The true statement about $\mathbf{y = ax^2 + bx + c}$ and $\mathbf{y = a(x - h)^2 + k}$ is that: the value of a remains the same

The original expression is given as:

$\mathbf{y = ax^2 + bx + c}$

He wants to rewrite it as:

$\mathbf{y = a(x - h)^2 + k}$

Expand the above equation

$\mathbf{y = a(x - h)(x - h) + k}$

Open brackets

$\mathbf{y = a(x^2 - hx - hx + h^2) + k}$

$\mathbf{y = a(x^2 - 2hx + h^2) + k}$

Remove bracket

$\mathbf{y = ax^2 - 2ahx + ah^2 + k}$

Compare the above equation to: $\mathbf{y = ax^2 + bx + c}$

$\mathbf{ax^2 = ax^2}$

$\mathbf{-2ahx = bx}$

$\mathbf{ah^2 + k = c}$

Analyzing the above equations

$\mathbf{ax^2 = ax^2}$

Divide both sides by $\mathbf{x^2}$

$\mathbf{a = a}$

The above equation means that, the value of a remains unchanged.

Hence, option (c) is true

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