Parametric equations describe a curve using a third variable, usually called a parameter.
Instead of writing \(y\) directly as a function of \(x\), both coordinates are written as
functions of \(t\):
\[
x=f(t),\qquad y=g(t).
\]
As \(t\) changes, the point \((x(t),y(t))\) moves and traces a curve.
1. What the parameter means
The parameter \(t\) often represents time, but it does not have to. It can simply be an
auxiliary variable that helps describe a curve.
\[
t_{\min}\le t\le t_{\max}.
\]
The interval controls which part of the curve is drawn.
2. Curve tracing
To graph a parametric curve, choose values of \(t\), compute \(x(t)\) and \(y(t)\), then plot
the resulting ordered pairs.
\[
t_1,t_2,t_3,\ldots
\]
\[
(x(t_1),y(t_1)),\ (x(t_2),y(t_2)),\ (x(t_3),y(t_3)),\ldots
\]
Connecting these points in order shows the direction of the curve.
3. Projectile motion example
A common parametric model is projectile motion. If horizontal velocity is \(v_x\), initial
vertical velocity is \(v_y\), initial height is \(h\), and gravity is \(g\), then:
\[
x=v_xt,
\]
\[
y=h+v_yt-\frac{1}{2}gt^2.
\]
The sample:
\[
x=3t,\qquad y=6t-5t^2
\]
represents a parabolic path.
4. Eliminating the parameter
Sometimes we can convert parametric equations to a Cartesian equation by eliminating \(t\).
For example:
\[
x=3t.
\]
Solve for \(t\):
\[
t=\frac{x}{3}.
\]
Substitute into:
\[
y=6t-5t^2.
\]
Then:
\[
y=6\left(\frac{x}{3}\right)-5\left(\frac{x}{3}\right)^2.
\]
Simplifying:
\[
y=2x-\frac{5}{9}x^2.
\]
5. Parametric lines
A line can be written parametrically as:
\[
x=x_0+at,
\]
\[
y=y_0+bt.
\]
If \(a\ne0\), solve:
\[
t=\frac{x-x_0}{a}.
\]
Substitute into \(y=y_0+bt\):
\[
y=y_0+b\left(\frac{x-x_0}{a}\right).
\]
6. Circles
A circle of radius \(r\) centered at the origin can be written as:
\[
x=r\cos t,
\]
\[
y=r\sin t.
\]
Since:
\[
\cos^2 t+\sin^2 t=1,
\]
the Cartesian equation is:
\[
x^2+y^2=r^2.
\]
7. Ellipses
An ellipse centered at the origin can be written as:
\[
x=a\cos t,
\]
\[
y=b\sin t.
\]
Divide by the corresponding scale factors:
\[
\frac{x}{a}=\cos t,\qquad \frac{y}{b}=\sin t.
\]
Then use \(\cos^2 t+\sin^2 t=1\):
\[
\frac{x^2}{a^2}+\frac{y^2}{b^2}=1.
\]
8. Spirals
A spiral can be modeled with:
\[
x=t\cos t,\qquad y=t\sin t.
\]
Here the distance from the origin increases as \(t\) increases. In polar form, this is related to:
\[
r=t.
\]
Many spirals do not have a simple Cartesian equation.
9. Lissajous curves
A Lissajous curve can be written as:
\[
x=\sin(at),
\]
\[
y=\sin(bt).
\]
These curves depend strongly on the ratio \(a:b\). They are often easier to study parametrically
than in Cartesian form.
10. Direction of motion
Parametric equations include direction. The point moves from:
\[
(x(t_{\min}),y(t_{\min}))
\]
toward:
\[
(x(t_{\max}),y(t_{\max})).
\]
This is different from a basic Cartesian equation, which often shows only the shape and not the tracing direction.
11. Velocity and speed
If \(t\) represents time, then the velocity vector is:
\[
\vec v(t)=\left(\frac{dx}{dt},\frac{dy}{dt}\right).
\]
The speed is the magnitude of this vector:
\[
|\vec v(t)|=\sqrt{\left(\frac{dx}{dt}\right)^2+\left(\frac{dy}{dt}\right)^2}.
\]
12. Arc length
The arc length of a smooth parametric curve from \(t=a\) to \(t=b\) is:
\[
L=\int_a^b \sqrt{\left(\frac{dx}{dt}\right)^2+\left(\frac{dy}{dt}\right)^2}\,dt.
\]
A graphical calculator usually approximates arc length by adding many small straight-line distances
between nearby sampled points.
13. Formula summary
14. Common mistakes
- Forgetting that \(x\) and \(y\) both depend on \(t\).
- Drawing points out of order and losing the direction of motion.
- Assuming every parametric curve has a simple Cartesian equation.
- Using too small a \(t\)-interval and missing part of the curve.
- Using too few sample points for a rapidly changing curve.
- Confusing the parameter \(t\) with the x-coordinate.
Key idea: parametric equations describe motion along a curve. The graph is made by computing
\((x(t),y(t))\) for many values of \(t\), and Cartesian conversion is possible only when \(t\)
can be eliminated cleanly.
