Parametric Equations and Graphs

Parametric Equations for Circles

Parametric equations allow us to represent curves by expressing the coordinates as functions of a parameter, usually denoted as t. For a circle, the standard parametric equations are derived from the trigonometric definitions of sine and cosine.

• General Form: For a circle with center at (h, k) and radius r:

• Example: For a circle with center at (-3, 2) and radius 4:

• Graph: The graph of these equations is a circle centered at (-3, 2) with radius 4. As t varies from 0 to , the point traces the entire circle once in the counterclockwise direction.

Parametric Equations for Ellipses

An ellipse can also be represented parametrically. The standard form for an ellipse centered at (h, k) with semi-major axis a and semi-minor axis b is:

• General Form:

• Example: For an ellipse with center at (-3, 2), vertices at (-3, 7) and (-3, -3), and other points at (1, 2) and (-7, 2):

• The vertices (-3, 7) and (-3, -3) are vertically aligned, so the major axis is vertical.

• The distance from the center to a vertex is , so (semi-major axis).

• The points (1, 2) and (-7, 2) are horizontally aligned with the center, so (semi-minor axis).

• Graph: The graph of these equations is an ellipse centered at (-3, 2), stretching 5 units above and below the center (major axis), and 4 units left and right (minor axis).

Summary Table: Parametric Equations for Circles and Ellipses

Key Points

• Parametric equations provide a way to describe curves using a parameter, often simplifying the process of graphing and analyzing conic sections.

• For circles and ellipses, the parameter t typically represents the angle in radians measured from the positive x-axis.

• Changing the values of h and k translates the center of the curve, while r, a, and b control the size and shape.