Overview
This lecture explains how to eliminate the parameter from vector-valued functions to find the Cartesian equation, using examples of lines and ellipses, and discusses how this aids in graphing.
Eliminating the Parameter: Linear Example
- Given x = 3t - 2 and y = 4t - 3.
- To eliminate t, solve for t: t = (x + 2)/3.
- Substitute into y: y = 4((x + 2)/3) - 3.
- Simplifies to y = (4/3)x - (1/3), a straight line.
- Slope is 4/3 and y-intercept is -1/3.
- Directionality: as t increases, x and y both increase, so the particle moves from bottom-left to top-right.
Cartesian Equations for Circles and Ellipses
- Standard circle: x² + y² = r², center at origin, radius r.
- Dividing by r² gives x²/r² + y²/r² = 1.
- For ellipses, use different denominators: x²/a² + y²/b² = 1.
- a is the semi-axis in the x-direction; b is the semi-axis in the y-direction.
- Extends to -a and -b in the negative x and y directions.
Eliminating the Parameter: Ellipse Example
- Given x = 4cos(t), y = -2sin(t).
- Square both sides: x² = 16cos²(t), y² = 4sin²(t).
- Divide: x²/16 + y²/4 = cos²(t) + sin²(t) = 1 (using trig identity).
- This matches the standard ellipse form: x²/16 + y²/4 = 1.
- The ellipse has semi-axes 4 (x-direction) and 2 (y-direction).
Key Terms & Definitions
- Parameter Elimination — The process of removing the parameter (e.g., t) from parametric equations to get a Cartesian equation.
- Ellipse — A curve defined by x²/a² + y²/b² = 1; a and b are the semi-major and semi-minor axes.
- Slope — The rate of change in y with respect to x in a line, calculated as rise over run.
Action Items / Next Steps
- Practice eliminating the parameter in other vector-valued functions.
- Review how to sketch graphs from resulting Cartesian equations.
- Prepare for a more challenging example in the next lecture.