Defining Parametric Equations (College Board AP® Calculus BC): Study Guide

Defining parametric equations

What are parametric equations?

• Parametric equations are an alternative way to represent the equation of a curve using a third variable called a parameter

  • The letter $t$ is often used for the parameter

  • The parametric equations are

    • $x=f\left(t\right)$

    • $y=g\left(t\right)$

• For example

  • $x=t-5$

  • $y=t^2+e^t$

• Sometimes a restricted range of $t$ values is given

  • e.g. where $0\le t\le 8$

How do I sketch parametric equations?

• You can sketch parametric curves using your calculator

  • Change the type of graph to "parametric"

• Parametric curves often have very interesting shapes

  • They can loop around and spiral etc.

• Parametric curves have a direction of flow

  • e.g. anticlockwise around a loop

  • The direction can be found by looking at the equations as $t$ increases

• You can find key features of the graph algebraically

  • The $y$-intercepts are when $x=0$

    • If $x=f\left(t\right)$ then solve $f\left(t\right)=0$

  • The $x$-intercepts are when $y=0$

    • If $y=g\left(t\right)$ then solve $g\left(t\right)=0$

How do I eliminate the parameter?

• Eliminating the parameter means rewriting a parametric curve as an equation in $x$ and $y$ only (no $t$)

  • To do this:

    • Make $t$ the subject of one of the equations

    • Substitute this into the other equation

    • Simplify if required

  • e.g. make $t$ the subject of $x=t-5$ and substitute it into $y=t^2+e^t$

    • This gives $y={\left(x+5\right)}^2+e^{x+5}$

    • There is no $t$

• You may need to convert a range of $t$ values into a range of $x$ or $y$ values

  • See the Worked Example

• It is not always possible to eliminate the parameter

  • e.g. $x=t+e^t$ and $y=\sin t+\ln t$

  • If it is possible, it is not always possible to make $y$ the subject

    • e.g. $y{e}^y=x+1$

How do I use trigonometric identities to eliminate the parameter?

• Another way to eliminate the parameter is by substituting parametric equations into a trigonometric identity

  • e.g. rearrange and substitute $x=2+\sin t$ and $y=4\cos t$ into the identity ${\sin }^{2}t+{\cos }^{2}t=1$

    • This gives the equation ${\left(x-2\right)}^2+{\left(\frac{y}{4}\right)}^2=1$

    • There is no $t$