Complete the square to identify the vertex of y=x2−16x+14

Question

.(1 point)
Responses

(8,14)
left parenthesis 8 comma 14 right parenthesis

(8,−50)
left parenthesis 8 comma negative 50 right parenthesis

(8,30)
left parenthesis 8 comma 30 right parenthesis

(8,78)

Answer 1

To complete the square for the quadratic expression \( y = x^2 - 16x + 14 \), we follow these steps:

1. Identify the coefficient of \( x \): Here, the coefficient of \( x \) is \(-16\).

2. Take half of this coefficient and square it: \[ \left(\frac{-16}{2}\right)^2 = (-8)^2 = 64 \]

3. Rewrite the quadratic expression: We add and subtract 64 in the equation. \[ y = x^2 - 16x + 64 - 64 + 14 \] This simplifies to: \[ y = (x - 8)^2 - 50 \]

4. Identify the vertex: The vertex form of a parabola is \( y = a(x - h)^2 + k \), where \((h, k)\) is the vertex. Here, we have: \[ y = (x - 8)^2 - 50 \] The vertex is \((8, -50)\).

Therefore, the correct response is:

\((8, −50)\)