Defining curves with parametric equations

Defining curves with parametric equations

We have focused a lot on Cartesian equations, so it is now time to focus on Parametric Equations. In this section, we will learn that parametric equations are two functions, x and y, which are in terms of t, or theta. We denote the variables to be parameters. Then we will learn how to sketch these parametric curves. After, we will analyze how to convert a parametric equation to a Cartesian equation. This is known as eliminating the parameter. Sadly, not all parametric equations can be converted to Cartesian in a nice way. This is especially true for parametric equations with sine and cosine. Therefore, we will introduce another way of eliminating the parameter, which involves using trigonometric identities.


Let xx and yy both be functions in terms of tt. Then we call them parametric equations where:
Each value of tt can determine a point (x,y)(x, y) that we can use to plot in the graph. Keep in mind that the parameter is not limited to tt. Sometimes we use the parameter θ\theta instead.

The main goal in this section is to learn how to sketch the curves, and to eliminate the parameter to find the Cartesian equation
  • 1.
    Defining Curves with Parametric Equations Overview:
    Sketching Parametric Curves

    Eliminating the parameter part 1

    Eliminating the parameter part 2

  • 2.
    Sketching Parametric Curves
    Sketch the following parametric curves using table of values and identify the direction of motion:
    x=t2t x=t^2-t

    x=cos(θ)x= \cos (\theta)
    y=sin(θ)y= \sin (\theta)

    x=sin(θ)x= \sin (\theta)
    y=sin(2θ)y= \sin (2\theta) where 0θπ20 \leq \theta \leq \frac{\pi}{2}

  • 3.
    Finding the Cartesian Equation of the Curve
    Eliminate the parameter and find the Cartesian equation of the following curves:
    x=t3+1 x=t^3+1

    x=ln(2t) x= \ln (2t)

  • 4.
    Find the Cartesian Equation of the Curve with Trigonometric Identities
    Eliminate the parameter θ\theta and find the Cartesian equation of the following curves:
    x=sin(2θ)x= \sin (2\theta)
    y=cos(2θ)y= \cos (2\theta)

    x=5sin(θ)x= 5\sin (\theta)
    y=3cos(θ)y= 3\cos (\theta)