# 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.

#### Lessons

Let $x$ and $y$ both be functions in terms of $t$. Then we call them parametric equations where:
$x=f(t)$
$x=g(t)$
Each value of $t$ can determine a point $(x, y)$ that we can use to plot in the graph. Keep in mind that the parameter is not limited to $t$. Sometimes we use the parameter $\theta$ 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
• Introduction
Defining Curves with Parametric Equations Overview:
a)
Sketching Parametric Curves

b)
Eliminating the parameter part 1

c)
Eliminating the parameter part 2

• 1.
Sketching Parametric Curves
Sketch the following parametric curves using table of values and identify the direction of motion:
a)
$x=t^2-t$
$y=t$

b)
$x= \cos (\theta)$
$y= \sin (\theta)$

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

• 2.
Finding the Cartesian Equation of the Curve
Eliminate the parameter and find the Cartesian equation of the following curves:
a)
$x=t^3+1$
$y=t+3$

b)
$x= \ln (2t)$
$y=e^t$

• 3.
Find the Cartesian Equation of the Curve with Trigonometric Identities
Eliminate the parameter $\theta$ and find the Cartesian equation of the following curves:
a)
$x= \sin (2\theta)$
$y= \cos (2\theta)$

b)
$x= 5\sin (\theta)$
$y= 3\cos (\theta)$