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

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

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

(8,78)

Answer 1

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

1. Identify and rearrange the quadratic terms: \[ y = x^2 - 16x + 14 \]

2. Complete the square:

   • Take the coefficient of \( x \), which is \(-16\), divide it by 2 to get \(-8\), and then square it to get \(64\).
   • Add and subtract \(64\) inside the equation: \[ y = (x^2 - 16x + 64) - 64 + 14 \]

3. Rewrite the quadratic as a perfect square: \[ y = (x - 8)^2 - 50 \]

4. Identify the vertex: The vertex form of a quadratic \( y = a(x-h)^2 + k \) shows that the vertex is at point \( (h, k) \). Here, \( h = 8 \) and \( k = -50 \), so the vertex is \( (8, -50) \).

Thus, the correct response is:

\((8, -50)\)