Find the focus, vertex, equation of the directrix and the axis of the parabola x `= y^(2) - 2y + 3 `.

Answer

To solve the problem of finding the focus, vertex, equation of the directrix, and the axis of the parabola given by the equation \( x = y^2 - 2y + 3 \), we will follow these steps:

### Step 1: Rewrite the equation in standard form
We start with the equation:
\[ x = y^2 - 2y + 3 \]
We can complete the square for the \( y \) terms:
\[ x = (y^2 - 2y + 1) + 3 - 1 \]
\[ x = (y - 1)^2 + 2 \]
Now, we have the equation in the form:
\[ x - 2 = (y - 1)^2 \]
This can be rewritten as:
\[ (y - 1)^2 = x - 2 \]

### Step 2: Identify the vertex
The standard form of a parabola that opens to the right is:
\[ (y - k)^2 = 4p(x - h) \]
From our equation, we can see that:
- \( h = 2 \)
- \( k = 1 \)
Thus, the vertex \( (h, k) \) is:
\[ \text{Vertex} = (2, 1) \]

### Step 3: Find the value of \( p \)
In the standard form, \( 4p = 1 \) (since the coefficient of \( (y - 1)^2 \) is 1). Therefore:
\[ p = \frac{1}{4} \]

### Step 4: Find the focus
The focus of the parabola can be found using the vertex and the value of \( p \). Since the parabola opens to the right, the focus is located at:
\[ (h + p, k) = \left(2 + \frac{1}{4}, 1\right) = \left(\frac{9}{4}, 1\right) \]

### Step 5: Find the equation of the directrix
The equation of the directrix for a parabola that opens to the right is given by:
\[ x = h - p \]
Substituting the values we found:
\[ x = 2 - \frac{1}{4} = \frac{8}{4} - \frac{1}{4} = \frac{7}{4} \]

### Step 6: Find the axis of the parabola
The axis of the parabola is the line that passes through the vertex and is parallel to the axis of symmetry. For this parabola, the axis is a horizontal line:
\[ y = k = 1 \]

### Summary of Results
- **Vertex**: \( (2, 1) \)
- **Focus**: \( \left(\frac{9}{4}, 1\right) \)
- **Equation of the Directrix**: \( x = \frac{7}{4} \)
- **Axis of the Parabola**: \( y = 1 \)