The vertex of the parabola `y^(2)-2x+8x-23=0` is

Options

A. (3,1)
B. (-3,)
C. (1,3)
D. (-3,-1)

Answer

To find the vertex of the parabola given by the equation \( y^2 - 2y - 8x - 23 = 0 \), we will follow these steps:

### Step 1: Rearrange the equation
Start by rearranging the equation to isolate the \( x \) terms on one side.

\[ y^2 - 2y - 23 = 8x \]

### Step 2: Complete the square for \( y \)
Next, we will complete the square for the \( y \) terms. The expression \( y^2 - 2y \) can be completed as follows:

1. Take the coefficient of \( y \) (which is -2), divide it by 2 to get -1, and then square it to get 1.
2. Add and subtract this square inside the equation.

\[ y^2 - 2y + 1 - 1 - 23 = 8x \]

This simplifies to:

\[ (y - 1)^2 - 24 = 8x \]

### Step 3: Rearrange to standard form
Now, we can rearrange this equation to express \( x \) in terms of \( y \):

\[ (y - 1)^2 = 8x + 24 \]

\[ (y - 1)^2 = 8(x + 3) \]

### Step 4: Identify the vertex
Now, we can identify the vertex from the standard form of the parabola \( (y - k)^2 = 4p(x - h) \), where \( (h, k) \) is the vertex.

From our equation \( (y - 1)^2 = 8(x + 3) \):
- \( h = -3 \)
- \( k = 1 \)

Thus, the vertex of the parabola is:

\[ (-3, 1) \]

### Final Answer
The vertex of the parabola \( y^2 - 2y - 8x - 23 = 0 \) is \( (-3, 1) \).

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