 # Parametric Equations of a Parabola Formula, Examples | How to find Parametric Form of Parabola?

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 y2 = 4ax

• Parametric Coordinates of the Parabola y2 = 4ax are (at2, 2at)
• Parametric Equations of Parabola y2 = 4ax are x = at2 and y = 2at

Standard Equation of Parabola y2 = -4ax

• Parametric Coordinates of the Parabola y2 = -4ax are (-at2, 2at)
• Parametric Equations of Parabola y2 = -4ax are x = -at2 and y = 2at

Standard Equation of Parabola x2 = 4ay

• Parametric Coordinates of the Parabola x2 = 4ay are (2at, at2)
• Parametric Equations of Parabola x2 = 4ay are x = 2at, y = at2

Standard Equation of Parabola x2 = -4ay

• Parametric Coordinates of the Parabola x2 = 4ay are (2at, -at2)
• Parametric Equations of Parabola x2 = 4ay are x = 2at, y = -at2

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

Parametric Equations of Parabola (y-k)2 = 4a(x-h) are x=h+at2, and y = k+2at

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

1. Write the Parametric Equations of the Parabola y2 = 16x?

Solution:

Given Equation is in the form of y2 = 4ax

On Comparing the terms we have the 4a = 16

a = 4

The formula for Parametric Equations of the given parabola is x = at2 and y = 2at

Substitute the value of a to get the parametric equations i.e. x = 4t2 and y = 2*4*t = 8t

Therefore, Parametric Equations of Parabola y2 = 16x are x= 4t2 and y = 8t

2. Write the Parametric Equations of Parabola x2 = 12y?

Solution:

Given Equation is in the form of x2 = 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 =  at2

Substitute the value of a to get the parametric equations i.e. x = 2*3*t and y = 3t2

Therefore, Parametric Equations of Parabola x2 = 12y are x = 6t and y = 3t2

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+at2, and y = k+2at

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

x = 2+2t2 and y = 3+2*2t

x = 2+2t2 and y = 3+4t

Therefore, Parametric Equations of the Parabola (y-3)2 =8(x-2) are x = 2+2t2 and y = 3+4t

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