BackStudy Notes: Parametric Equations (Section 10.1)
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10.1 Parametric Equation Notes
Parametric equations are a method of defining a curve by expressing the coordinates of the points on the curve as functions of a variable, called a parameter. This approach is especially useful for describing curves that cannot be represented as functions in the form or .
Definition: A parametric equation is a pair of equations of the form and , where is the parameter.
Graphing: The set of points as varies over an interval forms the graph of the parametric equations.
Applications: Parametric equations are used to describe the motion of objects, curves in the plane, and more complex geometric figures.
Example: If and , then the ordered pair is: is a point on the graph for each value of
Graphing Parametric Equations
To graph a parametric equation, compute several values of , find the corresponding pairs, and plot these points in the coordinate plane.
Example: Graph the parametric equations , .
For selected values of :
When : ,
When : ,
When : ,
Plot these points and connect them smoothly to visualize the curve.
Eliminating the Parameter
Sometimes, it is useful to eliminate the parameter to find a direct relationship between and .
Method: Solve one of the parametric equations for and substitute into the other equation.
Example: Given , :
Solve for in terms of :
Substitute into :
This gives the Cartesian equation relating and .
Parametrization of a Line
Any straight line can be represented parametrically. This is useful for describing lines in vector and calculus applications.
Given:
Parametric Form: Let , then
Example: For :
Alternatively, for a line passing through two points and :
where traces the segment between the points.
Example: For points and :
Parametric Equations for Circles
A circle centered at with radius can be described parametrically as:
where
Example: For a circle centered at with radius $5$:
Parametric Equations for Ellipses
An ellipse centered at with horizontal radius and vertical radius is given by:
Standard form:
Parametric equations:
where
Example: For :
Summary Table: Common Parametric Forms
Curve | Parametric Equations | Parameter Range |
|---|---|---|
Line through and |
| |
Circle centered at , radius |
| |
Ellipse centered at , axes , |
|
Additional info:
Parametric equations are foundational for calculus topics such as derivatives and integrals of vector-valued functions, and for describing motion in physics and engineering.
Graphing calculators and software (e.g., Desmos) are useful tools for visualizing parametric curves.
