Complete the square to identify the vertex of y=x^2−16x+14

Question

.(1 point)
Responses

(8,78)
left parenthesis 8 comma 78 right parenthesis

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

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

(8,30)

Answer 1

To identify the vertex of the quadratic function \( y = x^2 - 16x + 14 \) by completing the square, follow these steps:

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

2. Focus on the quadratic and linear terms \( x^2 - 16x \). To complete the square, take half of the coefficient of \( x \) (which is -16), square it, and add and subtract it inside the equation: \[ -16 / 2 = -8 \quad \text{and} \quad (-8)^2 = 64 \]

3. Rewrite the equation by adding and subtracting 64: \[ y = (x^2 - 16x + 64) + 14 - 64 \]

4. This simplifies to: \[ y = (x - 8)^2 - 50 \]

5. Now, we can identify the vertex from the completed square form \( y = (x - h)^2 + k \), where \( (h, k) \) is the vertex. Here, \( h = 8 \) and \( k = -50 \).

Thus, the vertex of the function is: \[ \boxed{(8, -50)} \]