Parametric Equations

In a rectangular coordinate system, you will recall, a point in the plane is represented by an ordered pair of number (x,y), where x and y equal the signed distance of the point from the y-axis and the x-axis respectively. In a polar coordinate system, we select a point, called the pole, and then a ray with vertex at the pole, called the polar axis. Comparing the rectangular and polar coordinate systems, we see that the origin in rectangular coordinates coincides with the pole in polar coordinates, and the positive x-axis in rectangular coordinates coincides with the polar axis in polar coordinates.

Parametric EquationsTo see the usefulness of this procedure, consider the path of an object that is propelled into the air at an angle of . If the initial velocity of the object is 48 feet per second, it can be shown that the object follows the parabolic path However, this equation does not tell the whole story. Although it does tell you where the object has been, it doesn’t tell you when the object was at a given point(x, y) on the path. To determine this time, you can introduce a third variable t, called a parameter. It is possible to write both x and y as functions of t to obtain the parametric equations.

From this set of equations you can determine that at time t=0, the object is at the point (0,0). Similarly, at time t=1, the object is at the point ( and so on.

Sketching a Plane CurveThe way to sketch a curve represented by a pair of parametric equations is to plot points in the xy-plane. Each set of coordinates (x,y) is determined from a value chosen for the parameter t. By plotting the resulting points in the order of increasing values of t, you trace the curve in a specific direction. This is called the orientation of the curve.

Sketch the curve given by the parametric equations.

𝑥=𝑡 2−4 𝑦=𝑡2−2≤𝑡≤3

Describe the orientation of the curve.

Eliminating the ParameterMany curves that are represented by sets of parametric equations have graphs that can also be represented by rectangular equations (in x and y). The process of finding the rectangular equations called eliminating the parameter

Parametricequations

Solve for t in One equation

SubstituteIn secondequation

Rectangularequation

𝑥=𝑡 2−4𝑦=

12𝑡

𝑡=2 𝑦 𝑥=(2 𝑦 )2−4 𝑥=4 𝑦 2−4

Identify the curve represented by the equations.

and

or

𝑦=

1

𝑥2−1

1𝑥2−1+1

¿

1−𝑥2

𝑥2

1𝑥2

∗ 𝑥2

𝑥2=1−𝑥2

Sketch the curve represented by and by eliminating the parameter

and

𝑠𝑖𝑛2𝜃+𝑐𝑜𝑠2𝜃=1

( 𝑥3 )2

+( 𝑦4 )2

=1

𝑥2

9+ 𝑦2

16=1

So we have an ellipse centered at (0,0), with vertex (0,4) and (0,-4) and minor axis of length 2b=6