Important Parabola Formulas

Q: What is maximum parabola?

Vertical parabolas give an important piece of information: When the parabola opens up, the vertex is the lowest point on the graph — called the minimum, or min. When the parabola opens down, the vertex is the highest point on the graph — called the maximum, or max.

Q: What moves the parabola left and right?

The parent function for a parabolic function is where is the center of the parabola. To shift the parabola left or right, the value of h changes. Since there is a negative sign in the parent function, a positive value moves the parabola to the left and a negative value moves it to the right.

Parabolas are one of the most fascinating shapes in mathematics. They appear in everyday life, from the trajectory of a ball in sports to the design of satellite dishes and headlights. Understanding the mathematical properties of parabolas can help us solve real-world problems and excel in competitive exams like JEE and NEET.

In this article, we’ll break down parabola formulas into simple concepts so that anyone can understand them. Whether you're a student preparing for exams or someone curious about mathematics, this guide is for you.

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What is a Parabola?

A parabola is a symmetrical curve formed when a cone is sliced by a plane parallel to its side. In simpler terms, it's the U-shaped graph you see in quadratic equations like $y = x^2$ $y$ $=$ $x$ $2$ .

Key Elements of a Parabola

Before diving into formulas, let’s understand some important terms:

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Focus: A fixed point inside the parabola.
Directrix: A fixed straight line outside the parabola.
Vertex: The point where the parabola changes direction. It's the midpoint between the focus and the directrix.
Axis of Symmetry: A line that divides the parabola into two symmetrical halves.
Latus Rectum: A line segment perpendicular to the axis of symmetry and passing through the focus.

Standard Equations of Parabolas

The equation of a parabola changes based on its orientation (whether it opens up, down, left, or right). Let’s look at the formulas for different cases:

1. Parabola Opening Up or Down

The standard equation is:

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y^2 = 4ax

$y$ $2$ $=$ $4$ $ax$

If $a > 0$ $a$ $>$ $0$ , the parabola opens to the right.
If $a < 0$ $a$ $<$ $0$ , the parabola opens to the left.

2. Parabola Opening Right or Left

The standard equation is:

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x^2 = 4ay

$x$ $2$ $=$ $4$ $ay$

If $a > 0$ $a$ $>$ $0$ , the parabola opens up.
If $a < 0$ $a$ $<$ $0$ , the parabola opens down.

Vertex Form of a Parabola

Sometimes, a parabola doesn’t sit neatly on the origin ( $0, 0$ $0$ $,$ $0$ ). For such cases, we use the vertex form of the equation:

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

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

Here:

$(h, k)$ $($ $h$ $,$ $k$ $)$ is the vertex of the parabola.
$a$ $a$ controls how "wide" or "narrow" the parabola is.

Key Formulas for a Parabola

To solve problems involving parabolas, you’ll need to know these important formulas:

1. Focus

For a parabola $y^2 = 4ax$ $y$ $2$ $=$ $4$ $ax$ , the focus is:

(a, 0)

$($ $a$ $,$ $0$ $)$

For $x^2 = 4ay$ $x$ $2$ $=$ $4$ $ay$ , the focus is:

(0, a)

$($ $0$ $,$ $a$ $)$

2. Directrix

The equation of the directrix is a straight line:

For $y^2 = 4ax$ $y$ $2$ $=$ $4$ $ax$ , the directrix is: $x = -a$ $x$ $=$ $-$ $a$
For $x^2 = 4ay$ $x$ $2$ $=$ $4$ $ay$ , the directrix is: $y = -a$ $y$ $=$ $-$ $a$

3. Axis of Symmetry

The axis of symmetry is:

$x = 0$ $x$ $=$ $0$ for $y^2 = 4ax$ $y$ $2$ $=$ $4$ $ax$ .
$y = 0$ $y$ $=$ $0$ for $x^2 = 4ay$ $x$ $2$ $=$ $4$ $ay$ .

4. Latus Rectum

The length of the latus rectum is always:

|4a|

$∣4$ $a$ $∣$

For example:

In $y^2 = 4ax$ $y$ $2$ $=$ $4$ $ax$ , the endpoints of the latus rectum are: $(a, 2a) \text{ and } (a, -2a)$ $($ $a$ $,$ $2$ $a$ $)$ $and$ $($ $a$ $,$ $-2$ $a$ $)$
In $x^2 = 4ay$ $x$ $2$ $=$ $4$ $ay$ , the endpoints are: $(2a, a) \text{ and } (-2a, a)$ $($ $2$ $a$ $,$ $a$ $)$ $and$ $($ $-2$ $a$ $,$ $a$ $)$

Deriving Parabola Equations

Let’s quickly derive the basic parabola equation $y^2 = 4ax$ $y$ $2$ $=$ $4$ $ax$ .

Step-by-Step Derivation:

Assume the focus is $(a, 0)$ $($ $a$ $,$ $0$ $)$ and the directrix is $x = -a$ $x$ $=$ $-$ $a$ .
The definition of a parabola states that any point $(x, y)$ $($ $x$ $,$ $y$ $)$ on the parabola is equidistant from the focus and the directrix.
Using the distance formula:
- Distance to focus: $\sqrt{(x - a)^2 + y^2}$ $($ $x$ $-$ $a$ $)$ $2$ $+$ $y$ $2$ $$
- Distance to directrix: $|x + a|$ $∣$ $x$ $+$ $a$ $∣$
Equating these distances: $\sqrt{(x - a)^2 + y^2} = |x + a|$ $($ $x$ $-$ $a$ $)$ $2$ $+$ $y$ $2$ $$ $=$ $∣$ $x$ $+$ $a$ $∣$
Squaring both sides: $(x - a)^2 + y^2 = (x + a)^2$ $($ $x$ $-$ $a$ $)$ $2$ $+$ $y$ $2$ $=$ $($ $x$ $+$ $a$ $)$ $2$
Expanding and simplifying: $y^2 = 4ax$ $y$ $2$ $=$ $4$ $ax$

This is the standard equation of a parabola.

Applications of Parabola Formulas

Understanding parabolas isn’t just theoretical. Here’s where they’re useful:

1. Physics and Engineering

Parabolas describe the trajectory of projectiles.
They are used in designing satellite dishes and telescopes.

2. Mathematics and Exams

Parabola questions are common in competitive exams like JEE.
Understanding their geometry helps in solving integration and area problems.

3. Real-Life Examples

Parabolic bridges, arches, and car headlights use the reflective property of parabolas.

Common Questions on Parabolas

Here are a few examples to test your understanding:

Example 1:

Find the focus and directrix of the parabola $y^2 = 12x$ $y$ $2$ $=$ $12$ $x$ .

Solution:

Compare with $y^2 = 4ax$ $y$ $2$ $=$ $4$ $ax$ .
Here, $4a = 12$ $4$ $a$ $=$ $12$ , so $a = 3$ $a$ $=$ $3$ .
Focus: $(3, 0)$ $($ $3$ $,$ $0$ $)$ .
Directrix: $x = -3$ $x$ $=$ $-3$ .

Example 2:

Find the equation of the parabola with vertex $(2, 3)$ $($ $2$ $,$ $3$ $)$ and opening upwards.

Solution:

Use the vertex form: $y = a(x - h)^2 + k$ $y$ $=$ $a$ $($ $x$ $-$ $h$ $)$ $2$ $+$ $k$ .
Since the vertex is $(2, 3)$ $($ $2$ $,$ $3$ $)$ , the equation is: $y = a(x - 2)^2 + 3$ $y$ $=$ $a$ $($ $x$ $-$ $2$ $)$ $2$ $+$ $3$

To find $a$ $a$ , we need an additional point. For now, this is the general equation.

Example 3:

Find the length of the latus rectum for $x^2 = -8y$ $x$ $2$ $=$ $-8$ $y$ .

Solution:

Compare with $x^2 = 4ay$ $x$ $2$ $=$ $4$ $ay$ .
Here, $4a = -8$ $4$ $a$ $=$ $-8$ , so $a = -2$ $a$ $=$ $-2$ .
Length of the latus rectum: $|4a| = 8$ $∣4$ $a$ $∣$ $=$ $8$ .

Tips to Remember Parabola Formulas

Understand Symmetry: Parabolas are always symmetric about their axis.
Identify Orientation: Look at the equation to decide the direction (up, down, left, or right).
Memorize Key Points: Focus, directrix, and vertex are the backbone of solving parabola problems.
Practice: Solve plenty of examples to solidify these concepts.

FAQs on Important Parabola Formulas

What is maximum parabola?

Vertical parabolas give an important piece of information: When the parabola opens up, the vertex is the lowest point on the graph — called the minimum, or min. When the parabola opens down, the vertex is the highest point on the graph — called the maximum, or max.

What moves the parabola left and right?

The parent function for a parabolic function is where is the center of the parabola. To shift the parabola left or right, the value of h changes. Since there is a negative sign in the parent function, a positive value moves the parabola to the left and a negative value moves it to the right.