AP Precalculus / Unit 4: Functions Involving Parameters, Vectors, and Matrices
4.7 Parametrization of Implicitly Defined Functions
Notes
Turn an implicit curve into parametric equations:
- Ellipse $\displaystyle \frac{x^2}{a^2}+\frac{y^2}{b^2}=1$: $x=a\cos t$, $y=b\sin t$.
- Hyperbola: $x=a\sec t$, $y=b\tan t$.
- A function $y=f(x)$: $x=t$, $y=f(t)$.
Practice
A parametrization of $\displaystyle \frac{x^2}{25}+\frac{y^2}{4}=1$:
Use $a=5$, $b=2$.
A parametrization of $y=x^3$:
Let $x=t$.
$x=\cos t$, $y=\sin t$ works because:
Pythagorean identity.
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