Parametric Equations
The Essentials
Parametric equations simplify an original equations with x and y to a more workable form. Parametric equations can also describe motion in space as functions of time y(t) and x(t). It essentially turns a regular position equation into a vector of functions of time.
Example
Example 1)
if you throw a ball with an initial velocity in the x direction of 3 \( \frac{m}{s} \). Because it of gravity, it experiences an acceleration of -9.81 \( \frac{m}{s^2} \). There is no resistance or losses, so the position equations can be denoted as...
To find the velocity of the ball we can take the derivative of f(t).
The acceleration is the second derivative of the same equation.
Example 2)
When given the parameterization of an equation, we can convert a regular equation to a vector system of time. If we have a function \( y=3x+4 \), and we are given that \( x=t^2 \) . Then we can plug x(t) into the y(x) function to find the parametric system of \( f(t)=< t^2,3t^2+4 > \)
Example 3)
Sometimes we are given a parametric equation, and it is helpful to convert it back to a regular function. This can be done by solving for t and then substituting it back into the equation. If we have a given parameterization of \( f(t)=< t^2+3t,t+6 > \), we can solve for t and find an equation.
Once we have solved for t, we can plug it into the x part of the parameterized equations to find x as a function of y.
Practice
Evaluate these expressions:
- Find the parameterization of the equation \( 9=x^2+y^2\)
- What is the equation that correlates with the parametric equation \( f(t)=< 6t,4t^3+2t > \)
Solutions:
- \( f(t)=< 3cos(x),3sin(x) > \)
- \( y=\frac{1}{54}x^3+\frac{1}{3}x \)