Graph curves defined by x(t) and y(t), animate a moving point, see direction arrows, and explore parameter elimination — all interactively.
Equations & Settings
x =
y =
t min
t max
Animation
t = —
Curve Summary
t range
—
Closed curve?
—
x range
—
y range
—
Start point (t=t_min)
—
End point (t=t_max)
—
Eliminated form
—
Direction Guide
Arrowheads along the curve show the direction of increasing t. The curve is traversed from the gold start dot to the coral end dot.
How to Use This Grapher
1
Choose a preset or enter your own equations. Type expressions using t as the variable. You can use any Math.* function: Math.sin(t), Math.cos(t), Math.PI, t**2, etc.
2
Set the t range and steps. The t range controls which portion of the curve is drawn. More steps give a smoother curve but slightly slower rendering.
3
Click "Graph Curve". The canvas auto-scales to fit all points. Arrowheads every ~10% show the direction of traversal (increasing t). The gold dot marks the start; coral marks the end.
4
Click "Animate". A moving gold dot traces the curve in real time, displaying the current t value and (x, y) coordinates. Use Pause/Resume to stop and continue.
5
Check the summary panel. See the x and y ranges, whether the curve is closed (start = end), and — for simple cases — the equivalent Cartesian equation after eliminating the parameter.
Parametric Graph
Parametric curve
Start (t = t_min)
End (t = t_max)
Direction arrows
𝚫 What Are Parametric Equations?
Instead of writing y = f(x), parametric equations express both x and y as functions of a third variable called the parameter (usually t, often representing time):
x = x(t) y = y(t) t ∈ [a, b]
Key advantages over y = f(x):
Can describe curves that loop or fold back on themselves (like a circle or cycloid)
Encodes the direction of motion — increasing t sweeps the curve in a specific direction
Describes motion along a path with a built-in notion of speed
Can represent curves that aren't functions of x (multiple y values for one x)
↔ Eliminating the Parameter
When x(t) is invertible (one-to-one), you can solve for t in terms of x and substitute into y(t) to get a Cartesian equation:
Linear: if x = a + bt, then t = (x−a)/b, substitute → y = y((x−a)/b)
Identity: if x = t, then y = y(x) directly
Trig: if x = cos t, y = sin t, use cos²t + sin²t = 1 → x² + y² = 1
cos²(t) + sin²(t) = 1 ⟹ x² + y² = 1 (unit circle)
When x(t) is not invertible (e.g. sinusoidal), elimination may be impossible or produce only an implicit relation.