Horizontal Parabola (opens left/right)

$(y - k)^2 = 4p(x - h)$

• Vertex: $(h, k)$
• Focus: $(h + p, k)$
• Directrix: $x = h - p$
• Opens right if $p > 0$, left if $p < 0$

What is $p$?

$p$ is the focal distance (distance from vertex to focus)

Converting Forms

From: $y = ax^2 + bx + c$

To conic form:

1. Complete the square
2. Identify $4p$ from coefficient

Example: $y = x^2 + 4x + 1$ $(x + 2)^2 = y + 3$ $(x + 2)^2 = 1(y + 3)$

So $4p = 1$, thus $p = \frac{1}{4}$

Key Properties

• Axis of symmetry passes through vertex
• Focus lies on axis of symmetry
• Distance from any point on parabola to focus = distance to directrix

Step 2: Identify vertex Vertex: $(-2, 3)$

Step 3: Find $p$ $4p = 4$, so $p = 1$

Step 4: Find focus (horizontal parabola) Focus: $(h + p, k) = (-2 + 1, 3) = (-1, 3)$

Step 5: Find directrix Directrix: $x = h - p = -2 - 1 = -3$

Answer: Vertex $(-2, 3)$, Focus $(-1, 3)$, Directrix $x = -3$