10.1 Parametric Equations Homework

Parametric Equations

Parametric equations express the coordinates of points on a curve as functions of a variable, called the parameter (often denoted as t).

This approach is especially useful for describing curves that cannot be represented as functions in the form x = f(t) or y = g(t).

• Parametric Form: A curve in the plane can be described by equations of the form:

• Parameter Interval: The variable t is restricted to a specific interval, which determines the portion of the curve being traced.

• Graphing: To graph a parametric curve, compute pairs for various values of t in the given interval.

1. Exponential Parametric Equations

• Given: , ,

• Procedure:

  1. Choose integer values of t in the interval [-2, 3].

  2. For each t, compute and .

  3. Plot the points and label each with its corresponding t value.

• Application: This method is useful for visualizing how the parameter traces the curve.

2. Converting Parametric Equations to Cartesian Form

To eliminate the parameter and find a Cartesian equation relating and :

• Step 1: Solve one of the parametric equations for t in terms of or .

• Step 2: Substitute this expression into the other equation.

Example: For , :

1. Solve for :

2. Substitute into :

Cartesian Equation:

3. Exponential Parametric Equations

• Given: ,

• Conversion:

1. Solve for :

Substitute into :

4. Cartesian Equation:

5. Trigonometric Parametric Equations

• Given: , ,

• Conversion:

  1. Recall

  2. So (since )

6. Cartesian Equation:

• ,

7. Parametric Equations for Circles and Ellipses

Circle: Standard Parametric Form

• General Equation: For a circle with center and radius :

• Example: Center , radius 4:

9. Ellipse: Standard Parametric Form

• General Equation: For an ellipse centered at with semi-axes (horizontal) and (vertical):

• Given: Center , vertices at and , other points at and .

• Analysis:

  • Vertical distance from center to vertex: (so )

  • Horizontal distance from center to : (so )

• Parametric Equations:

Summary Table: Common Parametric Forms

Key Points and Applications

• Parametric equations allow for flexible descriptions of curves, including those not representable as functions.

• Conversion to Cartesian form is often possible by eliminating the parameter.

• Graphing parametric curves involves plotting pairs for various values of and labeling them appropriately.

• Circles and ellipses have standard parametric forms, useful for both graphing and analysis.