Parametric Functions
Introduced| English |
|---|
| parametric function/ˌpærəˈmetrɪk ˈfʌŋkʃn/ |
| parametric equations/ˌpærəˈmetrɪk ɪˈkweɪʒnz/ |
| parameter/pəˈræmɪtə/ |
Following a moving point
- A bird's flight cannot be captured by "$y$ as a function of $x$" — it loops and doubles back.
- Instead, describe where the bird is at each moment: its $x$ and its $y$, each depending on time.
- That is the parametric idea: a third variable drives both coordinates.
- It frees us to draw any path, not just curves that pass the vertical-line test.
What a parametric function is
- A parametric function 参数方程函数 gives two rules: $x = x(t)$ and $y = y(t)$.
- These parametric equations 参数方程 share a common input $t$.
- For each value of $t$, you get one point $(x(t), y(t))$ on the curve.
- As $t$ runs through its interval, the point sweeps out the whole path.
A parametric function describes a curve using…
A parametric function gives $x$ and $y$ each as a function of a parameter $t$: the pair $(x(t), y(t))$ traces the curve.
For $x = t^2 - 3$ and $y = t$, what is the $x$-coordinate when $t = 2$?
$x = 2^2 - 3 = 4 - 3 = 1$; and $y = t = 2$, so the point is $(1, 2)$.
A path, not a function of x
- Because both coordinates depend on $t$, the curve can loop, cross itself, or double back.
- It need not be a function of $x$ — a full circle or a spiral is perfectly fine.
- This is the key advantage over ordinary $y = f(x)$ graphs.
- The same picture would be impossible to write as a single $y$ in terms of $x$.
A parametric curve can trace a shape that is not a function of $x$ (like a full circle).
Because both coordinates depend on $t$, the path can loop or double back — it need not pass the vertical-line test.
The independent variable $t$ that drives both $x$ and $y$ is called the ____.
The parameter $t$ (often time) determines a single point $(x(t), y(t))$ for each of its values.
Select all true statements about parametric functions.
A parametric curve need not be a function of $x$ — that is its advantage. The other three are correct.
The parameter
- The driving variable $t$ is the parameter 参数, and it often stands for time.
- Different values of $t$ place the point at different spots along the curve.
- The direction of increasing $t$ gives the path an orientation — the way it is traced.
- Restricting $t$ to an interval draws only part of the curve.
What parametric form allows
In parametric form, x and y each depend on a parameter t. Sort each statement.
Eliminating the parameter
- Sometimes you can solve one equation for $t$ and substitute to get a plain $y$–$x$ relation.
- For $x = t$, $y = t^2$, substituting gives $y = x^2$ — a familiar parabola.
- This recovers the shape but throws away the timing and direction.
- Keep the parametric form when the motion, not just the shape, matters.
A parametric curve carries more information than its shape alone: it also has a direction and timing. Two different parametrizations can trace the same shape at different speeds or in opposite directions — do not treat them as identical.
Plot $x = t^2 - 3$, $y = t$ for $t = -2, -1, 0, 1, 2$.
- Points: $(1, -2), (-2, -1), (-3, 0), (-2, 1), (1, 2)$.
- Joined in order, they trace a sideways parabola opening to the right.
- No vertical line rule applies — this is not a function of $x$.
A parametric function describes a curve with parametric equations $x = x(t)$ and $y = y(t)$, driven by a parameter $t$ (often time). Because both coordinates depend on $t$, the path can loop and need not be a function of $x$, and it carries a direction and timing.