9.1 Defining and Differentiating Parametric Equations

Part of Parametric Equations, Polar Coordinates, and Vector-Valued Functions in AP® Calculus BC. Read the opening section, then create a free account for the full interactive lesson, practice, and progress tracking.

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A parametric equation describes both x and y in terms of a third variable, usually t. Instead of writing y directly as a function of x, you write x=f(t) and y=g(t). This is useful when motion or geometry is easier to describe through a parameter, such as time. To find the slope of the curve, you use

\frac{dy}{dx}=\frac{\frac{dy}{dt}}{\frac{dx}{dt}}

as long as

{\frac{dx}{dt}\ne0}

. This tells you how fast y changes compared to x, even though both are changing with respect to t.

You should think of parametric differentiation as a chain-rule idea: since both coordinates depend on t, the derivative with respect to x comes from comparing the two rates. If ${\frac{dy}{dt}}$ is positive and ${\frac{dx}{dt}}$ is also positive, then the slope is positive. If they have opposite signs, the slope is negative. Parametric equations can trace curves in ways ordinary y=f(x) equations cannot, such as loops or paths that double back, which is why this representation is so important in BC calculus.

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