# How To Find Vertex, Focus and Directrix of a Parabola Equation?

Short mathematical tutorial that explains you on How to find vertex, focus and directrix of a parabola equation.

## Finding Vertex, Focus and Directrix Of A Parabola

**What is a Parabola?**

A **parabola**
is a curve where any point is at an equal distance from a fixed point
(called the focus), and a fixed straight line (called the directrix).

There are two form of Parabola Equation Standard Form and Vertex Form.

**Standard Form:**

**y = ax**^{2} + bx + c

**Vertex Form:**

**y = a(x - h)**^{2} + k

**The Vertex of the Parabola:**

The vertex is a point V(h,k) on the parabola. Where, h and k can be found using the formula,

**h = -b / 2a**

k = 4ac - b^{2} / 4a

**The Focus of the Parabola:**

The focus is the point that lies on the axis of the symmetry on the parabola at,

**F(h, k + p), **

with p = 1/4a.

**The Directrix of the Parabola:**

The directrix of the parabola is the horizontal line on the side of the vertex opposite of the focus. The directrix is given by the equation.

**y = k - p**

This short tutorial helps you learn how to find vertex, focus, and directrix of a parabola equation with an example using the formulas.

**Example:** Consider a parabolic equation of the standard form y = 3x^{2} + 12x + 1. Find the vertex, focus and directrix.

**Given:**

y = 3x^{2} + 12x + 1

**Solution:**

We know that, the standard form of parabola equation is,

y = ax^{2} + bx + c

From which we know,

a = 3

b = 12

c = 1

**Step 1:** Finding Vertex of the Parabola Equation

Vertex V = (h,k)

Applying the values in the formula, we get,

h = -b / 2a = -12 / 2(3) = -2

k = 4ac - b^{2} / 4a = 4(3x12) 2 / 4(3) = 0

**Vertex V(-2, 0)**

Step 2: Finding Focus of the Parabola Equation

F(h, k + p),

with p = 1/4a

Applying the values in the formula, we get,

p = 1 / 4(3) = 0.083

k + p = 0 + 0.083 = 0.083

**Focus F(-2, 0.083)**

Step 3: Finding Directrix of the Parabola Equation

Applying the values in the formula, we get,

y = k - p = 0 - 0.083 = -0.083

**y = -0.083**

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