Parametric Equations
Parametric Equations
- You can write the x and y coordinates of each point on a curve as functions of a third variable.
- This variable is called a parameter and is often represented by the letter \(t\).
- To convert from parametric form to cartesian form, we can find \(t\) in terms of \(x\) and the substitute this into the other parametric equation, eliminating \(t\).
Example 1
- Curve C has parametric equations:
- \(x=\ln(t+2)\)
- \(y=\frac{1}{t+1}\)
- Find the cartesian equation:
- \(t=e^x-2\)
- \(y=\frac{1}{e^x-1}\)
Example 2
- Curve C has parametric equations:
- \(x=\ln t\)
- \(y=t^2-2\)
- Find the cartesian equation:
- \(t=e^x\)
- \(y=e^2x-1\)
A curve C has parametric equations:
- \(x=2t-1\)
- \(y=4t-7+\frac{3}{t}\)
- \(t=\frac{x+1}{2}\)
- \(y=4\left( \frac{x+1}{2} \right)-7+\frac{6}{x+1}\)
- \(y=2x-5+\frac{6}{x+1}\)
- \(y=\frac{(2x-5)(x+1)+6}{x+1}\)
- \(y=\frac{2x^2-3x+1}{x+1}\)
Find the cartesian form of these parametric equations, including their domain and range.
- \(x=2t+1\), \(y=\frac{1}{t}\), \(t>0\)
\(y=\frac{2}{x-1}\)
Domain: \(x>1\)
Range: \(y>0\)
- \(x=\frac{1}{t-2}\), \(y=t^2\), \(t>2\)
\(y=\frac{1}{x^2}+4\)
Domain: \(x>0\)
Range: \(y\geq4\)