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

Question

𝑦
=
𝑥
2
−
16
𝑥
+
14
.(1 point)
Responses

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

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

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

(8,−50)

Answer 1

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

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

2. To complete the square for \( x^2 - 16x \), take half of the coefficient of \( x \) (which is \(-16\)), square it, and add and subtract that value inside the equation. Half of \(-16\) is \(-8\), and squaring it gives \(64\).

3. Rewrite the quadratic expression: \[ y = (x^2 - 16x + 64) - 64 + 14 \] \[ y = (x - 8)^2 - 50 \]

4. The completed square form is: \[ y = (x - 8)^2 - 50 \]

5. From this equation, it is clear that the vertex of the parabola (in the form \( y = (x - h)^2 + k \)) is at the point \( (h, k) \), where \( h = 8 \) and \( k = -50 \).

Thus, the vertex is: \[ (8, -50) \]

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