Complete the square to identify the vertex of y=x^2−16x+14.(1 point)

Question

Responses

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

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

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

(8,14)

Answer 1

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

1. Identify the coefficient of \( x \): The coefficient of \( x \) is \(-16\).

2. Take half of this coefficient and square it: \[ \left( \frac{-16}{2} \right)^2 = (-8)^2 = 64 \]

3. Add and subtract this square inside the equation: \[ y = x^2 - 16x + 64 - 64 + 14 \]

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

Now the equation is in the vertex form \( y = a(x-h)^2 + k \), where \( (h, k) \) is the vertex.

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

• \( h = 8 \)
• \( k = -50 \)

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

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