1. Home
  2. >Calculus
  3. >Parametric Equations and Polar Coordinates
Help

Defining curves with parametric equations

Everything You Need in One Place

Homework problems? Exam preparation? Trying to grasp a concept or just brushing up the basics? Our extensive help & practice library have got you covered.

Learn and Practice With Ease

Our proven video lessons ease you through problems quickly, and you get tonnes of friendly practice on questions that trip students up on tests and finals.

Instant and Unlimited Help

Our personalized learning platform enables you to instantly find the exact walkthrough to your specific type of question. Activate unlimited help now!

Get the most by viewing this topic in your current grade. Pick your course now.

?
Intros
Lessons
  1. Defining Curves with Parametric Equations Overview:
  2. Sketching Parametric Curves
  3. Eliminating the parameter part 1
  4. Eliminating the parameter part 2
?
Examples
Lessons
  1. Sketching Parametric Curves
    Sketch the following parametric curves using table of values and identify the direction of motion:
    1. x=t2−t x=t^2-t
      y=ty=t
    2. x=cos⁡(θ)x= \cos (\theta)
      y=sin⁡(θ)y= \sin (\theta)
    3. x=sin⁡(θ)x= \sin (\theta)
      y=sin⁡(2θ)y= \sin (2\theta) where 0≤θ≤π20 \leq \theta \leq \frac{\pi}{2}
  2. Finding the Cartesian Equation of the Curve
    Eliminate the parameter and find the Cartesian equation of the following curves:
    1. x=t3+1 x=t^3+1
      y=t+3y=t+3
    2. x=ln⁡(2t) x= \ln (2t)
      y=ety=e^t
  3. Find the Cartesian Equation of the Curve with Trigonometric Identities
    Eliminate the parameter θ\theta and find the Cartesian equation of the following curves:
    1. x=sin⁡(2θ)x= \sin (2\theta)
      y=cos⁡(2θ)y= \cos (2\theta)
    2. x=5sin⁡(θ)x= 5\sin (\theta)
      y=3cos⁡(θ)y= 3\cos (\theta)
Topic Notes
?
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:
x=f(t)x=f(t)
x=g(t)x=g(t)
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