Parametric Equations – AP Calculus BC

In a traditional Cartesian equation like $y = f(x)$, the curve is described by a direct relationship between $x$ and $y$. Parametric equations introduce a third variable $t$ (the parameter), allowing us to describe curves that are not functions, such as circles, loops, and curves that cross themselves.

🎯 In this section you will learn

What parametric equations are and how to interpret them
How to eliminate the parameter to find a Cartesian equation
How to sketch parametric curves by plotting points
How to find the direction of motion as $t$ increases

📌 What Are Parametric Equations?

A pair of parametric equations is written as:

$$ x = f(t), \quad y = g(t) $$ where $t$ is the parameter, often representing time

💡 Key Idea

As $t$ varies, the point $(x, y)$ traces out a curve in the plane. This allows us to represent motion, direction, and curves that aren't functions.

📊 Visualizing Parametric Curves

Interactive graph below. See how the point moves as $t$ increases:

📌 Eliminating the Parameter

Often we can eliminate $t$ to find a Cartesian equation relating $x$ and $y$ directly.

Example 1Eliminating the Parameter (Line)

Given $x = t + 1$, $y = 2t - 3$, eliminate $t$.

①

Solve for $t$

$t = x - 1$

②

Substitute into $y$

$y = 2(x - 1) - 3 = 2x - 5$

③

Result

The curve is the line $y = 2x - 5$.

Example 2Eliminating the Parameter (Parabola)

Given $x = t^2$, $y = t + 1$, eliminate $t$.

From $y = t + 1$, $t = y - 1$
Substitute into $x$: $x = (y - 1)^2$
This is a parabola opening to the right.

Example 3Eliminating with Trig (Circle)

Given $x = 3\cos t$, $y = 3\sin t$, eliminate $t$.

$\cos t = x/3$, $\sin t = y/3$
$\cos^2 t + \sin^2 t = 1$ → $(x/3)^2 + (y/3)^2 = 1$ → $x^2 + y^2 = 9$
The curve is a circle of radius $3$ centered at the origin.

Example 4Eliminating with Trig (Ellipse)

Given $x = 2\cos t$, $y = 5\sin t$, eliminate $t$.

$\cos t = x/2$, $\sin t = y/5$
$\cos^2 t + \sin^2 t = 1$ → $\frac{x^2}{4} + \frac{y^2}{25} = 1$
The curve is an ellipse.

📌 Sketching Parametric Curves

To sketch a parametric curve:

Choose several values of $t$ (especially where direction changes).
Compute the corresponding $(x, y)$ points.
Plot the points and connect them in order of increasing $t$.
Draw arrows to indicate the direction of motion as $t$ increases.

⚠️ Common Mistakes

Not checking the domain of $t$: The parameter may be restricted (e.g., $0 \le t \le 2\pi$).
Ignoring direction: Parametric curves have direction — always indicate it with arrows.
Assuming $t$ is always time: $t$ can be any parameter, not necessarily time.

🔍 Key Takeaways

Parametric equations: $x = f(t)$, $y = g(t)$.
Eliminate $t$ by solving for $t$ in one equation and substituting into the other.
Trigonometric parametric equations often describe circles and ellipses.
The direction of motion is from smaller $t$ to larger $t$.

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