뒤로Parametric Equations and Cartesian Conversion
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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
t | x | y |
|---|---|---|
0 | ||
1 | ||
2 |
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.
