Equation of Parabola

A parabola is a section of the right cone that is parallel to one side (a producing line) of the conic figure. Much the same as the circle, the parabola is also a quadratic relation, but different from the circle, either ‘A’ will be squared or ‘B’ will be squared, but never both. That said, a parabola is a set of all points M(A, B) in a plane in a way that the distance from M to a definite point F known as the focus is equivalent to the distance from M to a definite line known as the directrix as shown below in the graph.

Parabolic Equation

The general equation of a parabola is y = x² in which x-squared is a parabola.  Work up its side it becomes y² = x or mathematically expressed as y = \[\sqrt{x}\]

The Formula for Equation of a Parabola

Taken as known the focus (h, k) and the directrix y = mx+b, parabola equation is y−mx–by−mx–by - mx – b² / m²+1m²+1m² +1 = (x - h)² + (y - k)² .

Spot the Parabola at a Stroke

The shape of the parabola is what you see when you buy an ice cream cone and snip it off parallel to the side of the cone.

Usefulness of Parabola

Think of simply an arc and you will know what a parabola is. Now you will learn about parabolas and their standard form equations since these equations are quite frequently applicable in real life.

If there weren't any parabolas, we wouldn't have mobile and telecommunication services wherever we go. We wouldn't have the latest dish television facility either! Mobile phones and satellite television put satellites into execution in outer space that are parabolic-shaped. Thus, the parabola can be used for:

• Radar dishes or communication satellite

• Satellite dishes

• The reflector on torches and spotlights

• Focusing the sun's rays to make a hot spot

The Equation for Parabola in Standard Form

Let’s learn how to describe our parabola algebraically. Our equation relies on the Cartesian coordinate plane with the axis namely x and y. We basically have two different types of standard equations that differ depending on the direction of our parabola.

Write an equation for parabolas that open its way to either up or down

For such parabolas, the standard form equation is x–hx–hx – h² = 4p y–ky–ky – k

Here, the focus point is provided by (h, k + p).

These parabolas open on the y-axis, and thus the p-value is added to the y value of the vertex.

Write equation for parabolas that open its way to sideways

For such parabolas, the standard form equation is (y - k)² = 4p x–hx–hx – h T

Here, the focus point is provided by (h + p, k)

These open on the x-axis, and thus the p-value is then added to the x value of our vertex

That said, these parabolas are all the more same, just that the x and y are swapped.  The 4p in the standard form of the equation for a parabola is always with the section that is not squared. The alphabets h, k, and p stand for numbers that facilitate us with helpful information about the position of our parabola.

The Equation for Parabola in Vertex Form

The vertex of a parabola is the point at which the parabola passes through its axis of symmetry. In case the coefficient of the x² term is positive, then the vertex will be located at the lowest point on the graph, the point at the base of the “U”-shape. On the contrary, if the coefficient of the x² term is negative, the vertex will be located at the highest point on the graph, at the top of the “ U ”-shape.

The equation for a parabola in "vertex form" can also be written as

y=a (x−h) 2+k

Solved Examples

Example 1: For a parabola's equation y= 3x2 +12x−12. Find out its vertex?

Solution 1: Given that,y= 3x2 +12x−12

Here, m=3 and n=12.

Thus, the x -coordinate of the vertex comes out to be:−12 2 (3)= −2

Now, Substituting in the original equation to obtain the y -coordinate, we acquire:y=3 −2−2−2 2+12 −2−2−2 −12    =−24

Thus, the vertex of the parabola is placed at (−2, −24)

Example 2: For a parabola's equation y= 2(x-3)2+4. Find out the vertex of a parabola with the following equation? Which way does it open upwards or downwards?

Solution 2: Vertex comes to be = (-3, 4), and it opens upwards being that ‘a’ is positive.

A bit more generally:y² = 4ax

In which ‘a’ is the distance from the point of origin to the F(and from the origin to directrix too)Y integral line of axis y = ax + b

Now appears to be,ax² + bx + c making y equal to 0 and 1

Thus,

We have, a x² + bx+cbx+cb x + c = 1

Then, = ax²ax²a x²  + bx+cbx+cb x + c = 0

Hence, we get the standard parabolic form as y = a x² + bx + c

Fun Facts 

• In the standard form of the parabola equation, the “h” always accompanies the ‘x’ and the “k” is always with the ‘y’ axis.

• One best way to not forget the pairing is to think of an alphabetical table. just like y comes after x in the alphabetical table, similarly k comes after h

Important Terms to Understand and Remember

Students must understand and remember the below-given terms because these terms are useful for understanding the concept of the parabola.

1. Focus: The point (a, 0) is taken as the focus of a parabola.

2. Directrix: The directrix is a line drawn parallel to the y-axis and it passes through a point (-a,0). The directrix is perpendicular to the axis of the parabola. 

3. Focal Chord: The line passing through the focus of the parabola is known as the focal chord of the parabola. The focal chord divides the parabola at two distinct points.

4. Focal Distance: The focal distance is the distance of a point x1y1 on the parabola from the focus. The focal distance is equal to the perpendicular distance of this point from the directrix.

5. Latus Rectum: Latus Rectum is the focal chord that is perpendicular to the axis of the parabola and passes through the focus. The length of the latus rectum can be calculated as LL’ = 4a. The endpoints of the latus rectum are taken as (a, 2a), (a, -2a)

6. Eccentricity: Eccentricity is the ratio of the distance of a point from the focus to the distance of the point from the directrix. The eccentricity of a parabola is 1.