## Quick Reference

(of a curve)

A method of associating, with every value of a parameter *t* in some interval *I* (or some other subset of **R**), a point *P*(*t*) on the curve such that every point of the curve corresponds to some value of *t*. Often this is done by giving the *x*- and *y*-coordinates of *P* as functions of *t*, so that the coordinates of *P* may be written (*x*(*t*), *y*(*t*). The equations that give *x* and *y* as functions of *t* are parametric equations for the curve. For example, *x*=*at*^{2}, *y*=2*at* (*t* ∈ **R**) are parametric equations for the parabola *y*^{2}=4*ax*; and *x*=*a* cos θ, *y*=*b* sin θ (θ ∈ [0, 2π) are parametric equations for the ellipseThe gradient *dy*/*dx* of the curve at any point can be found, if *x*′(*t*) ≠ 0, from *dy*/*dx*=*y*′(*t*)/*x*′(*t*).

*Subjects:*
Mathematics.

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