Parametric Equations

What Are Parametric Equations?

Instead of expressing $y$ directly as a function of $x$, parametric equations define both $x$ and $y$ in terms of a third variable called a parameter (usually $t$):

$x = f(t), \quad y = g(t)$

This allows us to describe curves that fail the vertical line test, trace motion over time, and represent complex shapes naturally.

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Eliminating the Parameter

To convert from parametric to rectangular form:

1. Solve one equation for $t$
2. Substitute into the other

Example: $x = 2t + 1$, $y = t^2$

$t = \frac{x-1}{2} \implies y = \left(\frac{x-1}{2}\right)^2 = \frac{(x-1)^2}{4}$

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Common Parametric Curves

Line

$x = x_0 + at, \quad y = y_0 + bt$

• Direction vector: $\langle a, b \rangle$
• Passes through $(x_0, y_0)$

Circle

$x = h + r\cos t, \quad y = k + r\sin t, \quad 0 \leq t \leq 2\pi$

• Center $(h, k)$, radius $r$

Ellipse

$x = a\cos t, \quad y = b\sin t$

• Eliminates to $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$

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Derivatives with Parametric Equations

$\frac{dy}{dx} = \frac{dy/dt}{dx/dt} \quad \text{(where } dx/dt \neq 0\text{)}$

Horizontal tangent: $dy/dt = 0$ (and $dx/dt \neq 0$)

Vertical tangent: $dx/dt = 0$ (and $dy/dt \neq 0$)

Second Derivative

$\frac{d^2y}{dx^2} = \frac{\frac{d}{dt}\left[\frac{dy}{dx}\right]}{\frac{dx}{dt}}$

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Arc Length of Parametric Curves

$L = \int_a^b \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2}\, dt$

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Motion Applications

When $t$ represents time:

• Position: $(x(t), y(t))$
• Velocity vector: $\langle dx/dt, dy/dt \rangle$
• Speed:

AP Precalculus Tip: Parametric equations are a major topic in Unit 4. Be comfortable eliminating parameters, finding slopes, and interpreting motion.