Topic 4.1 Notes – Parametric Functions

What a Parametric Function Is

A parametric function in the plane has the form

$$f(t)=(x(t),y(t))$$

This means one input, $t$, produces one output point in $\mathbb R^2$, which just means an ordered pair $(x,y)$.

You’ll also see the same idea written as the pair of equations

$$x=x(t), \qquad y=y(t)$$

Those are equivalent. They are just two ways to show that both coordinates come from the same parameter $t$.

You can think of the parameter as feeding into a rule that outputs a point in the plane.

Parametric function as point-valued output

A few details matter a lot:

• $t$ is the parameter. It is the shared independent variable for both coordinates.
• At a specific value $t_i$, the point is
  $$f(t_i)=(x(t_i),y(t_i))$$
• The graph is drawn in the $xy$-plane. You do not put $t$ on an axis.
• Unlike $y=g(x)$, neither coordinate is the input to the other. Both depend on $t$.
• Each allowed $t$-value must give exactly one point.
• Different $t$-values can give the same point. That still counts as a function of $t$.

From Equations to a Table and Graph

This process is what you’ll do on quizzes most often.

1. Write the full analytical form, meaning $x(t)$, $y(t)$, and the domain of $t$.
2. Pick values of $t$ from that domain.
3. Evaluate both $x(t)$ and $y(t)$ at the same $t$-value.
4. Record rows as $t,\ x(t),\ y(t),\ (x(t),y(t))$.
5. Keep the rows in increasing $t$-order.
6. Plot the ordered pairs in the $xy$-plane.
7. Connect them in increasing $t$-order.
8. Add arrows to show direction.

If the coordinate functions are smooth, your sketch should be smooth too. A table gives sample points from a curve, not usually a broken line graph.

How the Domain Controls the Graph

The domain of $t$ tells you which part of the curve actually appears.

• If $a \le t \le b$, the graph starts at $f(a)$ and ends at $f(b)$. Both endpoints are included.
• If an endpoint is excluded, the graph may show an open endpoint there, if that point is only approached and never actually produced.
• If the domain is unbounded, the curve may have no start point, no end point, or neither.
• Restricting the domain can shorten the curve, isolate one piece, or stop retracing.

So the formulas may describe a larger geometric curve, but the allowed $t$-values decide how much you draw.

Reading Across Representations

You need to move easily among the three forms:

• Analytical: formulas $x(t)$, $y(t)$, plus domain
• Numerical: table of $t$-values and points
• Graphical: plotted curve traced in parameter order
• Verbal: where it starts, where it ends, and how it moves as $t$ increases

Example

$$f(t)=(t^2-1,t), \qquad -2 \le t \le 2$$

Sample table:

Plotting those points and following them in increasing $t$ gives a right-opening parabolic arc. As $t$ increases, it goes from lower right to the leftmost point, then to upper right.

It fails the vertical line test as $y$ versus $x$, but it is still a valid parametric function because each $t$ gives one point.

Common Mistakes on the Exam

• Pairing $x(t_1)$ with $y(t_2)$. Each row must use the same parameter value.
• Connecting points by increasing $x$ instead of increasing $t$.
• Forgetting arrows, endpoints, or open versus closed endpoint behavior.
• Ignoring a restricted domain and drawing too much of the curve.
• Using too few points and missing a turn or bend.
• Treating $t$ like one graph axis instead of graphing $(x(t),y(t))$.