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.