 x^2=(Y-2) fin vertex, directrix, axis of symmetry, an focus

The equation \( x^2 = y - 2 \) describes a parabola. To analyze its properties like the vertex, directrix, axis of symmetry, and focus, we should first rewrite the equation in a standard form.

### Step 1: Rewrite in Standard Form
The standard form of a parabola that opens either upwards or downwards is:

\[
(x - h)^2 = 4p(y - k)
\]

where \((h, k)\) is the vertex of the parabola, and \(p\) is the distance from the vertex to the focus or directrix.

Given equation:
\[
x^2 = y - 2
\]

This can be rewritten as:
\[
x^2 = 1 \cdot (y - 2)
\]

Here, we can compare it with the standard form \( (x - h)^2 = 4p(y - k) \):

- \( h = 0 \)
- \( k = 2 \)
- \( 4p = 1 \) (so \( p = \frac{1}{4} \))

### Step 2: Identify the Vertex
The vertex of the parabola is given by \((h, k)\), which is:

\[
\text{Vertex} = (0, 2)
\]

### Step 3: Find the Focus
The focus lies at a distance \( p \) from the vertex along the axis of symmetry. Since \( p = \frac{1}{4} \) and the parabola opens upwards (as \( x^2 \) is positive):

\[
\text{Focus} = \left(0, 2 + \frac{1}{4}\right) = \left(0, \frac{9}{4}\right)
\]

### Step 4: Find the Directrix
The directrix is a line that is \( p \) units away from the vertex, but in the opposite direction (downwards in this case):

\[
\text{Directrix} = y = 2 - \frac{1}{4} = \frac{7}{4}
\]

### Step 5: Find the Axis of Symmetry
The axis of symmetry is the vertical line that passes through the vertex:

\[
\text{Axis of Symmetry} = x = 0
\]

### Summary of Properties:
- **Vertex:** \( (0, 2) \)
- **Focus:** \( \left(0, \frac{9}{4}\right) \)
- **Directrix:** \( y = \frac{7}{4} \)
- **Axis of Symmetry:** \( x = 0 \)

Would you like more details or have any other questions?

### Related Questions:
1. How would these properties change if the parabola opened sideways?
2. Can you derive the equation for a parabola with a given focus and directrix?
3. What would be the length of the latus rectum of this parabola?
4. How does the graph of this parabola look compared to \( y = x^2 \)?
5. How do changes in the coefficient of \( x^2 \) affect the focus and directrix?

### Tip:
Remember that the sign and coefficient in front of \( x^2 \) or \( y^2 \) in a parabola's equation determine the direction in which it opens and how "wide" or "narrow" it is.

Explore the properties of the parabola described by the equation x^2 = y - 2, including its vertex, directrix, axis of symmetry, and focus. Learn how to determine these key features step-by-step with detailed explanations. Suitable for high school students studying parabolas in algebra.

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