Parabolas describe many natural phenomena like the motion of objects affected by gravity, increase or decrease in the population, amount of reagents in a chemical equation, etc. At times, you need to evaluate how variables change with respect to time. To track how variables change over time, you can put equations into Parametric Form.

Different Parametric Equations can be used to represent a Parabola. We have listed the simple and easiest way on How to find the Parametric Equations of a Parabola in the below modules. Refer to the Solved Examples on Parametric Equations of Parabola for a better understanding of the concept.

## Standard Forms of Parabola and their Parametric Equations

Let us discuss in detail the Parametric Coordinates of a Point on Standard Forms of Parabola and their Parametric Equations

Standard Equation of Parabola y^{2} = 4ax

- Parametric Coordinates of the Parabola y
^{2}= 4ax are (at^{2}, 2at) - Parametric Equations of Parabola y
^{2}= 4ax are x = at^{2}and y = 2at

Standard Equation of Parabola y^{2} = -4ax

- Parametric Coordinates of the Parabola y
^{2}= -4ax are (-at^{2}, 2at) - Parametric Equations of Parabola y
^{2}= -4ax are x = -at^{2}and y = 2at

Standard Equation of Parabola x^{2} = 4ay

- Parametric Coordinates of the Parabola x
^{2}= 4ay are (2at, at^{2}) - Parametric Equations of Parabola x
^{2}= 4ay are x = 2at, y = at^{2}

Standard Equation of Parabola x^{2} = -4ay

- Parametric Coordinates of the Parabola x
^{2}= 4ay are (2at, -at^{2}) - Parametric Equations of Parabola x
^{2}= 4ay are x = 2at, y = -at^{2}

Standard Equation of Parabola (y-k)^{2} = 4a(x-h)

Parametric Equations of Parabola (y-k)^{2} = 4a(x-h) are x=h+at^{2}, and y = k+2at

### Solved Examples on finding the Parametric Equations of a Parabola

**1. Write the Parametric Equations of the Parabola y ^{2} = 16x?**

Solution:

Given Equation is in the form of y^{2} = 4ax

On Comparing the terms we have the 4a = 16

a = 4

The formula for Parametric Equations of the given parabola is x = at^{2} and y = 2at

Substitute the value of a to get the parametric equations i.e. x = 4t^{2} and y = 2*4*t = 8t

Therefore, Parametric Equations of Parabola y^{2} = 16x are x= 4t^{2} and y = 8t

**2. Write the Parametric Equations of Parabola x ^{2} = 12y?**

Solution:

Given Equation is in the form of x^{2} = 4ay

On Comparing the terms we have the 4a = 12

a = 3

The formula for Parametric Equations of the given parabola is x = 2at, and y = at^{2}

Substitute the value of a to get the parametric equations i.e. x = 2*3*t and y = 3t^{2}

Therefore, Parametric Equations of Parabola x^{2} = 12y are x = 6t and y = 3t^{2}

**3. Write the Parametric Equations of the Parabola (y-3) ^{2} =8(x-2)?**

Solution:

Given Equation is in the form of (y-k)^{2} = 4a(x-2)

Comparing the two equations we have k = 3, h = 2 and 4a = 8 i.e. a =2

The Formula for Parametric Equations of Parabola (y-k)^{2} = 4a(x-h) are x=h+at^{2}, and y = k+2at

substitute the values of k, a in the formula and obtain the parametric equation

x = 2+2t^{2} and y = 3+2*2t

x = 2+2t^{2} and y = 3+4t

Therefore, Parametric Equations of the Parabola (y-3)^{2} =8(x-2) are x = 2+2t^{2} and y = 3+4t