The image contains mathematical questions related to parabolas, ellipses, and hyperbolas, asking to determine their key features such as vertices, foci, and directrices. I will now work through one of the equations and provide detailed solutions for each.

Let’s start with the first equation: $a. \ y = \frac{1}{4}x^2$ This is a standard form of a parabola.

Steps to solve:

1. Identify the type of equation: The given equation $y = \frac{1}{4}x^2$ is a parabola in the form $y = ax^2$, which opens upwards since $a > 0$.

2. Find the vertex: The vertex for any parabola in the form $y = ax^2$ is at the origin $(0, 0)$.

3. Find the focus: The focus of a parabola is given by the formula $\left(0, \frac{1}{4a}\right)$. Here, $a = \frac{1}{4}$, so the focus is: $\left(0, \frac{1}{4 \times \frac{1}{4}} = 1\right)$ Thus, the focus is $(0, 1)$.

4. Find the directrix: The directrix is the line $y = -\frac{1}{4a}$. Substituting $a = \frac{1}{4}$: $y = -\frac{1}{4 \times \frac{1}{4}} = -1$ So, the directrix is the line $y = -1$.

5. Axis of symmetry: The axis of symmetry for this parabola is the vertical line passing through the vertex, i.e., $x = 0$.

Summary for equation $y = \frac{1}{4}x^2$:

• Vertex: $(0, 0)$
• Focus: $(0, 1)$
• Directrix: $y = -1$
• Axis of symmetry: $x = 0$

Would you like me to proceed with solving the next equations step-by-step?