The equation given is:

$x = \frac{1}{4}(y + 2)^2$

This equation resembles the standard form of a parabola that opens horizontally:

$x = a(y - k)^2 + h$

where $(h, k)$ is the vertex, and $a$ determines the orientation and width of the parabola. In this case, $a = \frac{1}{4}$, $h = 0$, and $k = -2$.

1. Vertex

The vertex $(h, k)$ of the parabola is at: $(0, -2)$

2. Focus

The focus of a horizontally oriented parabola is found by the formula: $\text{Focus} = (h + \frac{1}{4a}, k)$ Since $a = \frac{1}{4}$, we have: $h + \frac{1}{4a} = 0 + \frac{1}{4 \times \frac{1}{4}} = 1$ Thus, the focus is at: $(1, -2)$

3. Axis of Symmetry

For a horizontally oriented parabola, the axis of symmetry is a horizontal line passing through $y = k$. Therefore, the axis of symmetry is: $y = -2$

4. Directrix

The directrix of the parabola is given by: $x = h - \frac{1}{4a}$ In this case: $h - \frac{1}{4a} = 0 - 1 = -1$ So, the directrix is: $x = -1$

5. General Form of the Equation

The equation provided is already in a standard form for a horizontal parabola. Converting it into a general form, $Ax^2 + By^2 + Cx + Dy + E = 0$, is unnecessary for the properties of conic sections but can be rewritten as: $4x = (y + 2)^2$ or $4x - y^2 - 4y - 4 = 0$ if expanded fully.

Summary of Key Properties

• Vertex: $(0, -2)$
• Focus: $(1, -2)$
• Axis of Symmetry: $y = -2$
• Directrix: $x = -1$
• General Form: $4x - y^2 - 4y - 4 = 0$

Would you like more detailed steps on how these were derived, or do you have any questions?

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Here are 5 questions for further exploration:

1. How does the value of $a$ affect the orientation and shape of the parabola?
2. What are the key differences between horizontal and vertical parabolas?
3. Can a parabola have more than one axis of symmetry? Why or why not?
4. How is the distance between the vertex and the focus related to the directrix?
5. What transformations would you apply to the general equation if the parabola was rotated?

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Tip: When dealing with parabolas, remember that the value of $a$ determines how "wide" or "narrow" the parabola opens, and the sign of $a$ determines its orientation.