Complete the square to identify the vertex of y=x2−16x+14 .(1 point)

Question

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 equation \( y = x^2 - 16x + 14 \), we will follow these steps:

1. Start with the equation:
   \( y = x^2 - 16x + 14 \)

2. Focus on the quadratic and linear terms:
   \( x^2 - 16x \)

3. To complete the square, take half of the coefficient of \( x \) (which is \(-16\)), square it, and add and subtract this value. Half of \(-16\) is \(-8\) and squaring it gives \(64\): \[ y = (x^2 - 16x + 64) - 64 + 14 \]

4. Now, rewrite the equation: \[ y = (x - 8)^2 - 50 \]

5. Now, we can identify the vertex from the completed square form \( y = (x - h)^2 + k \), where \((h, k)\) is the vertex. Here:

   • \(h = 8\)
   • \(k = -50\)

Thus, the vertex of the given quadratic is \((8, -50)\).

The correct answer is: \[ (8, -50) \]