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 = ax² + 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 - b² / 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 = 3x² + 12x + 1. Find the vertex, focus and directrix.
Given:
y = 3x² + 12x + 1

Solution:
We know that, the standard form of parabola equation is,
y = ax² + 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 - b² / 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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