Parametric Equations and Cartesian Conversion

Graphing Parametric Equations

Parametric equations express the coordinates of points on a curve as functions of a variable, typically t (the parameter). This approach is useful for describing curves that cannot be represented easily by a single function in Cartesian form.

• Definition: A parametric equation defines both x and y as functions of a third variable, usually t.

• Example: For problem 3, the parametric equations are:

• Graphing: To graph these equations, choose several values of t, compute the corresponding (x, y) pairs, and plot them. Label each point with its t value for clarity.

Sample Table of Values

Additional info: Students are instructed to use Desmos, a graphing calculator, to visualize the parametric curve and label points with their t-values.

Converting Parametric Equations to Cartesian Form

To convert parametric equations to a Cartesian equation, eliminate the parameter t to express y directly in terms of x.

• Step 1: Solve one of the parametric equations for t or for the shared expression.

• Step 2: Substitute into the other equation to eliminate t.

• Example for Problem 3: Given and : - Solve for from the second equation: - Substitute into the first equation: Cartesian Equation: Or equivalently,

Key Points

• Parametric equations are useful for representing curves where y is not a function of x, or for describing motion.

• Converting to Cartesian form can simplify analysis and graphing, but may lose information about the direction or speed of traversal.

Example Application

• Given , , the Cartesian form describes a straight line, but the parametric form shows how the point moves along the line as t increases.