Graphing Parametric Equations and Eliminating the Parameter
Introduction
- Focus on graphing parametric equations.
- Discussion on eliminating the parameter.
- Example parametric equations and their solutions.
Basic Example
- Equations:
- ( x = t + 2 )
- ( y = t^2 )
- ( t ) from (-2) to (2)
- Steps:
- Make a table with columns ( t, x, y ).
- Calculate ( x ) and ( y ) for each ( t ) value.
- Plot the ( (x,y) ) points ignoring ( t ).
- Draw arrows indicating increasing ( t ).
Further Example: Square Roots
- Equations:
- ( x = \sqrt{t} )
- ( y = 3t + 1 )
- Domain Change: ( t \geq 0 ).
- Choose values of ( t ): 0, 1, 4, 9.
- Calculate and plot ( x ) and ( y ) values.
Eliminating the Parameter
- Example:
- ( x = 2t - 4 ), ( y = 4t^2 ).
- Process:
- Solve for ( t ) in terms of ( x ).
- Substitute ( t ) into the second equation.
- Graph the resulting ( y = f(x) ).
Trigonometric Functions
- Example:
- ( x = 3 \sin t )
- ( y = 3 \cos t )
- ( t ) between 0 and ( 2\pi ).
- Use identity: ( \sin^2 t + \cos^2 t = 1 ).
- Graph results in a circle.
Graphing Ellipses
- Example:
- ( x = 3 + 2 \cos t )
- ( y = -1 + 2 \sin t )
- Process:
- Isolate ( \cos t ) and ( \sin t ).
- Use Pythagorean identity.
- Resulting equation defines an ellipse.
Exponential Functions
- Example:
- ( x = 2^t )
- ( y = 2^{-t} )
- Eliminate ( t ) using logarithms.
- Resulting ( y = \frac{1}{x} ) equation.
Adding a Parameter
- Transform rectangular equations back into parametric form.
- Example:
- Equation: ( y = 3x + 5 ).
- Set ( t = x ), then ( y = 3t + 5 ).
Practice Problems
- Practice turning equations like ( y = x^2 + 3 ) and ( y = x^3 + 8 ) into parametric form.
- Use substitution and algebraic manipulation.
Conclusion
- Understanding of graphing parametric equations.
- Techniques for eliminating parameters to form Cartesian equations.
- How to add parameters to equations.
This summary captures key learnings on parametric equations and their conversions. Use these notes for quick reference and study.