Complete the square to identify the vertex of y=x2−16x+14

Question

.(1 point)
Responses

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

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

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

(8,30)

Answer 1

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

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

2. Focus on completing the square for the \( x^2 - 16x \) part. To do this, take the coefficient of \( x \) (which is \(-16\)), divide it by 2 (which is \(-8\)), and then square it (which gives \( 64 \)).

3. Rewrite the equation, adding and subtracting \( 64 \): \[ y = (x^2 - 16x + 64) - 64 + 14 \] \[ y = (x - 8)^2 - 50 \]

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

Now, we can identify the vertex. The vertex form of a parabola is given by \( y = a(x - h)^2 + k \), where the vertex is \((h, k)\).

From our expression \( y = (x - 8)^2 - 50 \), we see that:

• \( h = 8 \)
• \( k = -50 \)

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

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