Find the foci , vertices, directrices and axes of following parabola . Also draw its rough sketch `y = x^(2) - 2x + 3 `

Answer

To solve the problem step by step, we will analyze the given parabola equation \( y = x^2 - 2x + 3 \) and find its foci, vertices, directrices, and axes.

### Step 1: Rewrite the equation in standard form
Start with the given equation:
\[ y = x^2 - 2x + 3 \]
We can complete the square for the quadratic expression in \( x \).

### Step 2: Completing the square
1. Rearrange the equation:
\[ y = (x^2 - 2x) + 3 \]
2. Complete the square for \( x^2 - 2x \):
\[ x^2 - 2x = (x - 1)^2 - 1 \]
3. Substitute back into the equation:
\[ y = (x - 1)^2 - 1 + 3 \]
\[ y = (x - 1)^2 + 2 \]

### Step 3: Identify the standard form of the parabola
The equation can now be written as:
\[ (y - 2) = (x - 1)^2 \]
This is in the form \( (y - k) = a(x - h)^2 \), where:
- Vertex \( (h, k) = (1, 2) \)
- Here, \( a = 1 \).

### Step 4: Determine the parameters
For a parabola of the form \( (y - k) = a(x - h)^2 \):
- The vertex is at \( (h, k) = (1, 2) \).
- The distance \( a \) from the vertex to the focus and directrix is given by \( a = \frac{1}{4p} \), where \( p \) is the distance from the vertex to the focus.

Since \( a = 1 \), we have:
\[ 4p = 1 \implies p = \frac{1}{4} \]

### Step 5: Find the focus and directrix
1. **Focus**: The focus is located \( p \) units above the vertex (since the parabola opens upwards):
\[ \text{Focus} = \left(1, 2 + \frac{1}{4}\right) = \left(1, \frac{9}{4}\right) \]
2. **Directrix**: The directrix is a horizontal line \( p \) units below the vertex:
\[ \text{Directrix} = y = 2 - \frac{1}{4} = \frac{7}{4} \]

### Step 6: Identify the axis of symmetry
The axis of symmetry is the vertical line through the vertex:
\[ \text{Axis} = x = 1 \]

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

### Step 7: Rough sketch of the parabola
1. Plot the vertex at \( (1, 2) \).
2. Mark the focus at \( (1, \frac{9}{4}) \).
3. Draw the directrix line at \( y = \frac{7}{4} \).
4. Sketch the parabola opening upwards from the vertex.