Assignment 10

Parametric Equations

by Emily Bradley

For various a, b investigate

for

We know how to represent a graph by a single equation involving two variables. We can also represent curves in a plane in which three variables are used. We have x, y, and a third variable t, called a parameter. By writing both x and y as functions of t, we have the Parametric Equations.

The parameter can represent time, or in the following examples an angle.

This is the basic parametric equation for a circle with radius one. We can sketch the curve ourselves by eliminating the parameter and finding the corresponding rectangular equation.

cos t = x and sin t = y

Make use of the identity sin²t + cos²t = 1 to form an equation involving only x and y with substitution.

x² + y² = 1

We can see that the graph is of a circle centered at (0, 0), with radius 1.

Ellipse a smooth closed curve, symmetric about its center.

We can sketch the curve of

x = 4 cos t and y = 2 sin t, 0 ≤ t ≤ 2pi

again by eliminating the parameter and finging the corresponding rectangular equation. Solve for cos t and sin t.

cos t = x/4 and sin t = y/2

Using the identity sin²t + cos²t = 1 to form an equation involving only x and y with substitution.

(x/4)² + (y/2)² = 1

x²/16 + y²/4 = 1

We can see that the graph is of an ellipse centered at (0, 0), with vertices at (0, 2) , (0, -2) and (4, 0) , (-4, 0).

For the curve of

x = 2 cos t and y = 4 sin t, 0 ≤ t ≤ 2pi

Solve for cos t and sin t.

cos t = x/2 and sin t = y/4

Using the identity sin²t + cos²t = 1 to form an equation involving only x and y with substitution.

(x/2)² + (y/4)² = 1

x²/4 + y²/16 = 1

We can see that the graph is of an ellipse centered at (0, 0), with vertices at (0, 4) , (0, -4) and (2, 0) , (-2, 0).

Ellipse parametric form in conical position with varying a

By varying n from 0 to 4, the vertices along the X axis vary with respect to n.

The rectangular equation is

x²/n² + y² = 1

This graph is an ellipse centered at (0, 0) with vertices at (0, 1) , (0, -1) and (n, 0) , (-n, 0).

Ellipse parametric form in conical position with varying b

By varying n from 0 to 4, the vertices along the Y axis vary with respect to n.

The rectanglular equation is

x² + y²/n² = 1

This graph is an ellipse centered at (0, 0) with vertices at (0, n) , (0, -n) and (1, 0) , (-1, 0).

Circle parametric form with varying a and b

By varying n from 0 to 4, the vertices along the X and Y axis vary with respect to n.

The rectanglular equation is

x²/n² + y²/n² = 1

Thus the graph is of a circle with radius n centered at (0, 0) with vertices at (0, n) , (0, -n) and (n, 0) , (-n, 0).