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

Question

Responses

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

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

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

(8,30)

Answer 1

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

1. Identify the coefficients: We have \( a = 1 \), \( b = -16 \), and \( c = 14 \).

2. Complete the square:

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

\[ y = x^2 - 16x + 64 - 64 + 14 \]

3. Rewrite the equation: \[ y = (x - 8)^2 - 50 \]

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

Thus, the vertex of the parabola given by the equation \( y = x^2 - 16x + 14 \) is:

\[ (8, -50) \]

So the correct response is:

(8, −50)