Question
Complete the square to identify the vertex of y=x2−16x+14 .(1 point)
Responses
(8,14)
left parenthesis 8 comma 14 right parenthesis
(8,−50)
left parenthesis 8 comma negative 50 right parenthesis
(8,30)
left parenthesis 8 comma 30 right parenthesis
(8,78)
Answers
Answered by
GPT-4o mini
To complete the square for the quadratic equation \( y = x^2 - 16x + 14 \), follow these steps:
1. **Identify the coefficient of \( x \)**: Here, the coefficient of \( x \) is \(-16\).
2. **Take half of that coefficient and square it**:
\[
\left( \frac{-16}{2} \right)^2 = (-8)^2 = 64
\]
3. **Rewrite the equation, incorporating this value**:
To complete the square, we rewrite the equation like this:
\[
y = x^2 - 16x + 64 - 64 + 14
\]
Simplifying that, we have:
\[
y = (x - 8)^2 - 50
\]
4. **Identify the vertex**:
The vertex form of a parabola is given by \( y = a(x - h)^2 + k \), where \( (h, k) \) is the vertex. From our completed square, we can see:
\[
h = 8 \quad \text{and} \quad k = -50
\]
Therefore, the vertex is \( (8, -50) \).
The correct response is:
\[
\boxed{(8, -50)}
\]
1. **Identify the coefficient of \( x \)**: Here, the coefficient of \( x \) is \(-16\).
2. **Take half of that coefficient and square it**:
\[
\left( \frac{-16}{2} \right)^2 = (-8)^2 = 64
\]
3. **Rewrite the equation, incorporating this value**:
To complete the square, we rewrite the equation like this:
\[
y = x^2 - 16x + 64 - 64 + 14
\]
Simplifying that, we have:
\[
y = (x - 8)^2 - 50
\]
4. **Identify the vertex**:
The vertex form of a parabola is given by \( y = a(x - h)^2 + k \), where \( (h, k) \) is the vertex. From our completed square, we can see:
\[
h = 8 \quad \text{and} \quad k = -50
\]
Therefore, the vertex is \( (8, -50) \).
The correct response is:
\[
\boxed{(8, -50)}
\]