A parametric curve writes both coordinates as functions of a shared parameter t: x=x(t), y=y(t). As t runs through its domain, the point (x(t),y(t)) traces a path with a built-in direction of travel. A single equation y=f(x) has no way to encode that direction at all.
A road on a map and a hiker's timestamped GPS log of walking that road describe the same path, but they're not the same object. The map just shows where the road goes. The log also records which direction the hiker walked, how fast, and whether they doubled back over their own earlier trail. The parametrization is the GPS log; the curve you'd draw connecting the dots is just the map.
A parametric curve is the path traced by a point whose coordinates, x=x(t) and y=y(t), are both written as functions of a shared parameter t. As t runs through its domain, the point (x(t),y(t)) sweeps out a path. Unlike a single equation y=f(x), the parametrization also records a direction of travel. Two different parametrizations can trace the identical set of points while disagreeing completely about direction, speed, or how many times a given point gets revisited. The TRACE is a fact about a set of points. The parametrization is a fact about a specific pair of functions of t that produce that set. Reading a table of (x(t),y(t)) at several t-values is still the most reliable way to see the shape and the direction at once. Do this before attempting any algebraic elimination of t.
Watch direction, not just shape
Press play, or drag the scrubber — notice the curve is retraced from x=-1 up through the origin and back out, never doubling back on itself here.
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A curve that crosses its own path
Scrub t across the whole range — find the two different t-values that land on the exact same point.
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Worked example
Reading direction from a parametrization
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Describe how the point (x(t),y(t)) = (t,\ t^2-2) moves as t increases from -2 to 2.
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2eliminating t here is easy too. Since x=t, substitute directly: y = x^2 - 2
Build a short table: t=-2,-1,0,1,2 give the points (-2,2), (-1,-1), (0,-2), (1,-1), (2,2).
eliminating t here is easy too. Since x=t, substitute directly: y = x^2 - 2
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Reading the table in order of increasing t, the point moves left to right along a parabola, descending to the vertex (0,-2) and then climbing back up. It never reverses direction, since x=t increases the entire time.
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\boxed{\text{The point traces } y=x^2-2 \text{ left to right, once, with no self-intersection.}}
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Where this goes next
Chapter 2: Vector-valued functions and motion
The very next chapter renames this idea \mathbf{r}(t) = \langle x(t), y(t), z(t)\rangle and starts differentiating and integrating it component by component. Velocity, acceleration, and arc length in space are all built directly on the table-and-trace habit reviewed here.
Quick review
Practice from earlier coursework, pulled in for this topic.
From College Algebra
A hiking app logs a trail as timestamped (latitude, longitude) pairs. Why would sorting those points by longitude before drawing the trail be a mistake, on a trail that loops back westward at some point?
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GPS coordinates are just numbers and sort perfectly well — the issue is that sorting by the wrong quantity (longitude, an output) scrambles the story that timestamp order (the parameter) preserves.
This is exactly the x-order versus t-order trap from this section — a looping trail is a real-world case where longitude, like x in x=t^2-4t, is not monotonic, so sorting by it fails.
There's no alphabetical concept for coordinates — the correct ordering principle is chronological (parameter) order, not any alphabetic or numeric sort of the outputs.
Why
A trail that loops back west has repeating or reversing longitude values, exactly like x=t^2-4t repeating x-values — so longitude alone can't define a unique order. Only the timestamp, playing the role of the parameter t, faithfully reconstructs which point came before which.
Think about which quantity here is playing the role of the parameter t, and which is playing the role of x(t).
From Calculus II — Integral Calculus & Infinite Series
For x(t)=4-t, y(t)=t^2, find the y-coordinate of the point where t=3.
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Why
y(3)=3^2=9 (and x(3)=4-3=1, giving the point (1,9)).
Correct answer
Also accepted:
Why
From College Algebra
Which of these parametrizations has x(t) always increasing (safe to connect left-to-right in t-order)?
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x=t^2-4t falls, bottoms out at t=2, then rises again — it is not monotonic.
x=3\cos t oscillates between -3 and 3 forever — it repeatedly rises and falls, never monotonic over all t.
x=t^2 falls as t climbs from -\infty to 0, then rises from 0 to \infty — not monotonic over all real t.
Why
x=t-1 has slope 1 in t — it climbs steadily for every real t, with no falling or turning-around, so connecting its points in t-order always agrees with connecting them in x-order.
Check each x(t) for whether it ever falls, turns around, or repeats a value as t ranges over all reals.