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

Question

A) (8, 78)
B) (8, 30)
C) (8, 14)
D) (8, -50)

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 original equation: \[ y = x^2 - 16x + 14 \]

2. Focus on the quadratic and linear terms: \[ y = (x^2 - 16x) + 14 \]

3. Complete the square:

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

4. Rewrite the equation: \[ y = ((x - 8)^2 - 64) + 14 \] Simplifying it gives you: \[ y = (x - 8)^2 - 50 \]

5. Identify the vertex: The vertex form of the equation is \( y = a(x - h)^2 + k \), where \( (h, k) \) is the vertex. From \( y = (x - 8)^2 - 50 \), we see:

   • \( h = 8 \)
   • \( k = -50 \)

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

The correct answer is: D) (8, -50).