The Central Idea
Suppose a point moves through the plane according to
\[ x=t \]and
\[ y=t^2. \]For each value of \(t\), we obtain one point:
\[ t=-2\Rightarrow(-2,4), \] \[ t=0\Rightarrow(0,0), \] \[ t=3\Rightarrow(3,9). \]The parameter \(t\) determines both coordinates.
Together,
\[ \boxed{x=t,\qquad y=t^2} \]form a parametric representation of the parabola
\[ y=x^2. \]What Is a Parameter?
In a parametric representation, a parameter is an independent input used to generate one or more dependent quantities.
For a plane curve, we often write
\[ x=x(t), \qquad y=y(t). \]The parameter \(t\) determines the coordinate pair
\[ (x(t),y(t)). \]Parameter Versus Model Parameter
The word parameter appears in more than one role in mathematics.
In a family such as
\[ y=mx+b, \]\(m\) and \(b\) may be fixed model parameters.
In a parametric curve such as
\[ x(t),\qquad y(t), \]the parameter \(t\) is the active input that moves through the curve.
The context tells us which meaning is intended.
Parametric Form
A common two-dimensional parametric representation is
\[ \boxed{ x=f(t), \qquad y=g(t) } \]for \(t\) in some parameter domain.
Each allowed \(t\)-value gives one point in the plane.
One Input, Two Outputs
A parametric curve can be viewed as one function with a vector-valued output:
\[ \mathbf{r}(t) = \langle x(t),y(t)\rangle. \]The parameter \(t\) is the input.
The output is a point in the plane.
Parametric Representation as a Function
For example,
\[ \mathbf{r}(t) = \langle t,t^2\rangle \]is a function
\[ \mathbb{R}\to\mathbb{R}^2. \]Each real input \(t\) produces exactly one ordered pair.
The Parameter Need Not Be Time
The letter \(t\) often suggests time, especially in motion problems.
But the parameter can represent:
- time;
- angle;
- distance traveled;
- an index;
- or simply an abstract input.
The letter itself does not determine the meaning.
Parametric Representation of a Line
Consider
\[ x=2+t, \] \[ y=5+3t. \]At
\[ t=0, \]the point is
\[ (2,5). \]As \(t\) increases by \(1\), \(x\) increases by \(1\) and \(y\) increases by \(3\).
This traces a line.
Eliminating the Parameter
Sometimes we can remove the parameter and recover an equation relating \(x\) and \(y\).
From
\[ x=2+t, \]solve for \(t\):
\[ t=x-2. \]Substitute into
\[ y=5+3t: \] \[ y=5+3(x-2). \]So
\[ \boxed{y=3x-1}. \]Parametric and Explicit Forms Can Describe the Same Curve
The line
\[ y=3x-1 \]and the parametric equations
\[ x=2+t, \qquad y=5+3t \]describe the same geometric set of points when \(t\in\mathbb{R}\).
They emphasize different structures.
Why Use Parametric Form?
Parametric representations are useful because they can:
- describe motion naturally;
- represent curves that are not functions \(y=f(x)\);
- show direction along a curve;
- control how fast a curve is traced;
- represent restricted portions of curves;
- extend naturally to three or more dimensions.
A Circle Is Naturally Parametric
The circle
\[ x^2+y^2=r^2 \]does not define \(y\) as a single function of \(x\) over the whole circle.
But it has the parametric representation
\[ \boxed{ x=r\cos t, \qquad y=r\sin t }. \]Why the Circle Representation Works
Substitute
\[ x=r\cos t \]and
\[ y=r\sin t \]into
\[ x^2+y^2. \]Then
\[ x^2+y^2 = r^2\cos^2t+r^2\sin^2t. \]Factor:
\[ r^2(\cos^2t+\sin^2t). \]Using
\[ \cos^2t+\sin^2t=1, \]we obtain
\[ x^2+y^2=r^2. \]Tracing the Unit Circle
For
\[ x=\cos t, \qquad y=\sin t, \]the point starts at
\[ (1,0) \]when
\[ t=0. \]As \(t\) increases, the point moves counterclockwise around the unit circle.
Orientation
A parametric representation does more than identify the set of points on a curve.
It can also describe the orientation: the direction in which the curve is traced as the parameter increases.
Same Curve, Opposite Orientation
Compare
\[ x=\cos t, \qquad y=\sin t \]with
\[ x=\cos t, \qquad y=-\sin t. \]Both trace the unit circle.
But they travel around it in opposite directions as \(t\) increases.
Parameter Domain Controls Which Part Is Traced
For
\[ x=\cos t, \qquad y=\sin t, \]if
\[ 0\le t\le2\pi, \]the entire circle is traced once.
If
\[ 0\le t\le\pi, \]only the upper semicircle is traced.
Parameter Restrictions Matter
The same equations can represent different geometric objects depending on the parameter domain.
So a complete parametric representation includes both:
- the coordinate functions;
- the allowed parameter values.
Repeated Tracing
If
\[ x=\cos t, \qquad y=\sin t \]with
\[ 0\le t\le4\pi, \]the unit circle is traced twice.
The geometric curve is unchanged, but the parametrization records a different traversal.
Different Parameters Can Reach the Same Point
A parametric representation need not be injective.
For the circle,
\[ t=0 \]and
\[ t=2\pi \]both produce
\[ (1,0). \]The parameter function is still valid because each parameter input produces exactly one point.
Repeated Points Are Allowed
Different parameter values may produce the same coordinate point.
This is no different from an ordinary function where different inputs may share an output.
The curve parametrization need not be one-to-one.
Motion Interpretation
Suppose
\[ x(t)=3t, \qquad y(t)=2t+1. \]If \(t\) is measured in seconds, then the point's position changes over time.
At each time \(t\), the ordered pair
\[ (x(t),y(t)) \]gives the position.
Position as a Vector Function
The same motion can be written
\[ \mathbf{r}(t) = \langle 3t,2t+1\rangle. \]This is a vector-valued function of time.
Parametric notation and vector notation are two closely related ways to describe motion.
Velocity Preview
If
\[ \mathbf{r}(t) = \langle x(t),y(t)\rangle, \]then later in calculus, velocity is obtained by differentiating the coordinate functions:
\[ \mathbf{v}(t) = \langle x'(t),y'(t)\rangle. \]Parametric representations therefore connect naturally to motion and calculus.
Different Parametrizations Can Trace the Same Curve
Consider
\[ x=t,\qquad y=t^2 \]and
\[ x=2s,\qquad y=4s^2. \]Eliminating either parameter gives
\[ y=x^2. \]Both represent the same parabola.
The Speed of Tracing Can Differ
Even when two parametrizations trace the same curve in the same direction, they may move through it at different rates.
The geometric path and the parametrization are related but not identical concepts.
Example of Different Tracing Speeds
Compare
\[ x=t,\qquad y=t^2 \]with
\[ x=2t,\qquad y=4t^2. \]Both trace the parabola
\[ y=x^2. \]But equal changes in \(t\) produce different changes in position.
Eliminating a Parameter by Substitution
Suppose
\[ x=t+1, \qquad y=t^2. \]From the first equation,
\[ t=x-1. \]Substitute:
\[ y=(x-1)^2. \]So the Cartesian equation is
\[ \boxed{y=(x-1)^2}. \]Eliminating a Parameter with an Identity
For
\[ x=3\cos t, \qquad y=2\sin t, \]we have
\[ \frac{x}{3}=\cos t \]and
\[ \frac{y}{2}=\sin t. \]Square and add:
\[ \frac{x^2}{9}+\frac{y^2}{4} = \cos^2t+\sin^2t = 1. \]Thus,
\[ \boxed{\frac{x^2}{9}+\frac{y^2}{4}=1}. \]Parametric Representation of an Ellipse
The ellipse
\[ \frac{x^2}{a^2}+\frac{y^2}{b^2}=1 \]has a natural parametrization:
\[ \boxed{ x=a\cos t, \qquad y=b\sin t }. \]Parametric Representation of a Horizontal Line
The line
\[ y=4 \]can be parametrized by
\[ x=t, \qquad y=4. \]The parameter controls horizontal position while the vertical coordinate remains constant.
Parametric Representation of a Vertical Line
The vertical line
\[ x=3 \]cannot be written globally as \(y=f(x)\).
But it has the simple parametrization
\[ \boxed{ x=3, \qquad y=t }. \]This is one of the clearest advantages of parametric form.
Parametric Curves Can Fail the Vertical-Line Test
A parametric curve does not need to define \(y\) as a function of \(x\).
The parameter \(t\) is the actual input.
As long as each \(t\) produces one point, the parametrization itself is a valid function
\[ t\mapsto(x(t),y(t)). \]Relation to Implicit Curves
An implicit equation such as
\[ x^2+y^2=25 \]describes a relation geometrically.
A parametrization such as
\[ x=5\cos t, \qquad y=5\sin t \]describes how points on that relation are generated.
Explicit, Implicit, and Parametric Forms
| Form | Example |
|---|---|
| Explicit | \(y=x^2\) |
| Implicit | \(y-x^2=0\) |
| Parametric | \(x=t,\ y=t^2\) |
These can describe the same geometric relationship from different viewpoints.
Parametric Representation in Three Dimensions
A space curve can be written
\[ x=x(t), \qquad y=y(t), \qquad z=z(t). \]Equivalently,
\[ \mathbf{r}(t) = \langle x(t),y(t),z(t)\rangle. \]Three-Dimensional Example
Consider
\[ x=\cos t, \] \[ y=\sin t, \] \[ z=t. \]The \(x\)- and \(y\)-coordinates move around a circle while \(z\) increases.
The resulting curve is a helix.
Parametric Surfaces Preview
Two parameters can generate a surface:
\[ \mathbf{r}(u,v) = \langle x(u,v),y(u,v),z(u,v)\rangle. \]Now the input is an ordered pair
\[ (u,v), \]and the output is a point in three-dimensional space.
Parameters and Dimensions
The number of parameters often reflects the intrinsic dimension of the object being generated.
Very roughly:
- one parameter can trace a curve;
- two parameters can sweep out a surface;
- three parameters can describe a volume or higher-dimensional region.
Discrete Parametric Representations
A parameter does not need to vary continuously.
For integer \(n\),
\[ x=n, \qquad y=n^2 \]generates only the discrete points
\[ (n,n^2). \]The parameter domain determines whether the representation is continuous or discrete.
Parameter Domain and Curve Extent
Suppose
\[ x=t, \qquad y=t^2, \]but restrict
\[ -2\le t\le3. \]Then only the portion of the parabola with
\[ -2\le x\le3 \]is traced.
Recovering the Parameter
Sometimes one coordinate directly reveals the parameter.
If
\[ x=t+4, \]then
\[ t=x-4. \]This makes elimination straightforward.
The Parameter May Not Be Recoverable Uniquely
For
\[ x=\cos t, \]many values of \(t\) produce the same \(x\)-coordinate.
This is one reason eliminating the parameter can lose information about direction and repeated tracing.
Elimination Can Lose Orientation Information
The Cartesian equation
\[ x^2+y^2=1 \]describes the unit circle.
It does not tell us whether the circle is traversed clockwise or counterclockwise.
The parametrization does.
Elimination Can Lose Timing Information
If two parametrizations trace the same path at different speeds, eliminating the parameter produces the same Cartesian equation.
The geometric equation remembers the path but not how the path was traversed.
Parametric Representations as Models
Suppose a projectile is modeled by
\[ x(t)=v_xt, \] \[ y(t)=h_0+v_yt-\frac12gt^2. \]Time \(t\) determines both horizontal and vertical position.
This is a natural parametric model of motion.
Why Parametric Motion Is Natural
In motion, neither \(x\) nor \(y\) is necessarily the most natural independent variable.
Both depend on time.
Parametric form keeps time as the common input and lets each coordinate respond to it.
Different Coordinates Can Have Different Units
In ordinary geometric curves, \(x\) and \(y\) often share length units.
But more general parametric systems can generate several quantities with different units.
Dimensional consistency still applies to each coordinate function individually.
Worked Example: Generate Points
Let
\[ x=t+1, \qquad y=2t-3. \]For \(t=0\):
\[ (x,y)=(1,-3). \]For \(t=2\):
\[ (x,y)=(3,1). \]For \(t=-1\):
\[ (x,y)=(0,-5). \]Worked Example: Eliminate the Parameter
Using
\[ x=t+1, \qquad y=2t-3, \]solve the first equation:
\[ t=x-1. \]Then
\[ y=2(x-1)-3. \]So
\[ \boxed{y=2x-5}. \]Worked Example: Circle
Let
\[ x=4\cos t, \qquad y=4\sin t. \]Then
\[ \frac{x^2}{16}+\frac{y^2}{16} = \cos^2t+\sin^2t = 1. \]Therefore,
\[ \boxed{x^2+y^2=16}. \]Worked Example: Restricted Circle
Let
\[ x=4\cos t, \qquad y=4\sin t \]with
\[ 0\le t\le\pi. \]The curve is the upper semicircle
\[ x^2+y^2=16, \qquad y\ge0. \]Worked Example: Vertical Line
Consider
\[ x=2, \qquad y=t. \]As \(t\) ranges over all real numbers, the point moves through every point with \(x=2\).
Thus, the parametrization describes the vertical line
\[ \boxed{x=2}. \]Worked Example: A Parabola with Restricted Domain
Let
\[ x=t, \qquad y=t^2 \]for
\[ t\ge0. \]Then only the right half of the parabola
\[ y=x^2 \]is traced:
\[ x\ge0. \]A Reliable Parametric-Evaluation Workflow
- Choose or identify the parameter value.
- Substitute it into every coordinate function.
- Form the ordered point from the resulting coordinates.
- Interpret direction or time if the parameter has physical meaning.
A Reliable Parameter-Elimination Workflow
- Look for one equation that can be solved easily for the parameter.
- Substitute that expression into the other coordinate equation.
- Or use an identity when trigonometric functions are involved.
- Simplify to an equation in \(x\) and \(y\).
- Carry over any restrictions implied by the parameter domain.
- Remember that orientation and repeated tracing may be lost.
A Reliable Curve-Interpretation Workflow
- Identify the parameter domain.
- Generate several strategically chosen points.
- Determine the direction of travel as the parameter increases.
- Eliminate the parameter when useful.
- Check whether the entire implicit curve or only part of it is traced.
- Check whether any points are traced more than once.
Common Errors
Treating \(t\) as One of the Coordinates
The parameter generates the coordinates; it is not automatically an axis coordinate.
Forgetting the Parameter Domain
The same equations can trace a full curve, a partial curve, or the same curve repeatedly depending on the allowed parameter values.
Assuming Every Parametric Curve Is \(y=f(x)\)
Parametric curves can include circles, vertical lines, and other relations that fail the vertical-line test.
Losing Restrictions When Eliminating the Parameter
The Cartesian equation may describe more points than the original parametrization actually traces.
Assuming One Curve Has Only One Parametrization
The same geometric path can have infinitely many parametrizations.
Ignoring Orientation
Reversing or changing the parameter can change the direction of traversal without changing the geometric curve.
Confusing a Parametric Input with a Fixed Model Parameter
The active curve parameter varies; fixed model parameters select which curve or model is being used.
Practice
1. Generate a Point
Let
\[ x=t+2, \qquad y=3t. \]Find the point when
\[ t=4. \]Check Answer
\[ x=4+2=6, \] \[ y=3(4)=12. \]Answer:
\[ \boxed{(6,12)}. \]2. Eliminate the Parameter
Let
\[ x=t-1, \qquad y=2t+5. \]Eliminate \(t\).
Check Answer
\[ t=x+1. \]Then
\[ y=2(x+1)+5. \]So
\[ \boxed{y=2x+7}. \]3. Circle Parametrization
What curve is represented by
\[ x=3\cos t, \qquad y=3\sin t? \]Check Answer
\[ x^2+y^2=9. \]It is the circle of radius \(3\) centered at the origin.
4. Restricted Parameter
For
\[ x=\cos t, \qquad y=\sin t, \]what part of the unit circle is traced when
\[ 0\le t\le\pi? \]Check Answer
The upper semicircle, including both endpoints.
5. Vertical Line
Give a parametric representation of
\[ x=5. \]Check Answer
One possible answer is
\[ \boxed{x=5,\qquad y=t,\qquad t\in\mathbb{R}}. \]6. Parabola
Eliminate the parameter from
\[ x=2t, \qquad y=4t^2. \]Check Answer
\[ t=\frac{x}{2}. \]Then
\[ y=4\left(\frac{x}{2}\right)^2=x^2. \]Answer:
\[ \boxed{y=x^2}. \]7. Same Point Twice
For
\[ x=\cos t, \qquad y=\sin t, \]do \(t=0\) and \(t=2\pi\) produce the same point?
Check Answer
Yes.
\[ (\cos0,\sin0)=(1,0) \]and
\[ (\cos2\pi,\sin2\pi)=(1,0). \]8. Vector Form
Write
\[ x=2t+1, \qquad y=t^2 \]as a vector-valued function.
Check Answer
\[ \boxed{\mathbf{r}(t)=\langle 2t+1,t^2\rangle}. \]Key Ideas
- A parametric representation uses one or more shared parameters to generate dependent quantities.
- In the plane, \(x=x(t)\) and \(y=y(t)\) define a function from the parameter domain into \(\mathbb{R}^2\).
- The parameter need not represent time, though time is a common interpretation.
- Parametric form can represent curves that are not functions \(y=f(x)\).
- The parameter domain determines which part of a curve is traced and how many times it is traced.
- Parametrizations carry orientation and traversal information that ordinary Cartesian equations may lose.
- Different parametrizations can describe the same geometric curve.
- Eliminating the parameter can produce an explicit or implicit equation relating the coordinates.
- Restrictions from the parameter domain must be preserved after elimination.
- Vector-valued notation provides a compact way to write parametric curves.
- One parameter naturally describes curves; several parameters can describe higher-dimensional objects.
- Parametric representations connect function theory to geometry, motion, vectors, and calculus.
Where This Leads
Parametric representations provide a flexible way to describe paths, curves, motion, and multidimensional relationships without forcing one coordinate to depend directly on another.
They lead naturally into vector-valued functions, conic sections, projectile motion, polar and parametric curves, three-dimensional geometry, velocity and acceleration, arc length, surface parametrization, and multivariable calculus.