Assignment 10

Parametric Equations

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).