Parametric Equations

Unit 10.1 of Math 152

• This isn’t in the curriculum but i wasn’t taught this so self study :)

Curves Defined by Parametric Equations

• Imagine a particle moves long curve c

  • 

    • It is impossible to create an equastion with just f(x) because it fails VLT.

      • But the x and y- coordinates of the particle are functions of time t
        • therefore we can write $x=f(t),and,y=g(t)$.

          • This is a convenient way of describing a curve

Parametric Equations

• Using the def, we can suppose that x and y are both given as functions of t (a parameter), by the equation

  • $x=f(t),y=g(t)$

    • this is called parametric equations. Each value of t determines a point (x,y), which we can plot in a coordinate plane. As t varies, the point $(x,y)=(f(t),g(t))$ varies and traces out a curve.

Examples

• EX 1.

  • Sketch and identify the curve defined by the parametric equations

    • $x=t^2-t$ and $y=t+1$
      • each value of t gives a point on the curve
        • using a table we can get an outline
          • 
            • now we can trace
              • t=-2 is the starting point and you would graph it sequentialy as its defined by the parameter.
              • 

Eliminating the Parameter

• you can algebraically remove the parameter from the equation by solving for t then solving for y

  • Ex.

    • $x=2t-1$ and $y=\frac{1}{2}t+1$
      • we can solve for t
        • $t=\frac{x+1}{2}$ and $t=2y-2$
          • from there we can make an equation by canceling out t
            • $t=\frac{x+1}{2}=2y-2$
              • there we can solve for y
                • $y=t=\frac{x+1}{4}+1$