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

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

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

(8,−50)

Answer 1

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

1. Identify the coefficients: The coefficient of \( x^2 \) is 1 (which we'll keep), the coefficient of \( x \) is -16, and the constant is 14.

2. Take half of the coefficient of \( x \): Half of -16 is -8.

3. Square it: \((-8)^2 = 64\).

4. Rewrite the equation: \[ y = x^2 - 16x + 64 - 64 + 14 \] This can be reorganized as: \[ y = (x - 8)^2 - 50 \]

5. Identify the vertex: The vertex form of a parabola \( y = a(x - h)^2 + k \) shows that the vertex is at the point \((h, k)\). In this case, \( h = 8 \) and \( k = -50 \).

Thus, the vertex of the quadratic \( y = x^2 - 16x + 14 \) is \( (8, -50) \).

Therefore, the correct response is: \[ (8, -50) \]