The Central Idea
In ordinary Cartesian form, a curve might be described by a direct relationship between \(x\) and \(y\).
In parametric form, both coordinates are written in terms of a third variable:
\[ \boxed{x=f(t),\qquad y=g(t)}. \]The variable \(t\) is called the parameter.
As \(t\) changes, the point
\[ (x,y)=(f(t),g(t)) \]moves through the coordinate plane and traces a curve.
What Is a Parameter?
A parameter is an auxiliary variable used to generate coordinates.
For example,
\[ x=t,\qquad y=t^2 \]assigns one ordered pair to each allowed value of \(t\).
If
\[ t=-2,-1,0,1,2, \]then the corresponding points are
\[ (-2,4),\ (-1,1),\ (0,0),\ (1,1),\ (2,4). \]A Parameter Need Not Represent Time
The letter \(t\) is common because parameters often represent time, but this is not required.
A parameter could represent:
- time;
- angle;
- distance traveled;
- an arbitrary real number;
- another quantity convenient for describing the curve.
Letters such as \(s\), \(u\), and \(\theta\) are also commonly used.
Parametric Equations Generate Ordered Pairs
Suppose
\[ x=2t+1, \qquad y=3-t. \]At
\[ t=0, \]the point is
\[ (1,3). \]At
\[ t=2, \]the point is
\[ (5,1). \]The parameter coordinates the two expressions so that each value of \(t\) produces one point on the curve.
Tracing a Parametric Curve
To understand a parametric curve, calculate several values of the parameter and observe the resulting points.
For
\[ x=t,\qquad y=t^2, \]| \(t\) | \(x\) | \(y\) |
|---|---|---|
| \(-2\) | \(-2\) | \(4\) |
| \(-1\) | \(-1\) | \(1\) |
| \(0\) | \(0\) | \(0\) |
| \(1\) | \(1\) | \(1\) |
| \(2\) | \(2\) | \(4\) |
These points trace the parabola \(y=x^2\).
Orientation
Parametric equations contain information that an ordinary Cartesian equation may not show: orientation.
Orientation tells the direction in which a curve is traced as the parameter increases.
For
\[ x=t,\qquad y=t^2, \]increasing \(t\) moves the point from the upper-left portion of the parabola down toward the vertex and then upward along the right-hand portion.
Same Curve, Different Orientation
Consider
\[ x=t,\qquad y=t^2 \]and
\[ x=-t,\qquad y=t^2. \]Both generate the same Cartesian parabola
\[ y=x^2. \]However, they trace it in opposite directions as \(t\) increases.
Parametric descriptions therefore carry more information than the set of points alone.
Eliminating the Parameter
Sometimes a parametric description can be converted into a Cartesian equation by eliminating the parameter.
Suppose
\[ x=2t+1, \qquad y=5t-3. \]Solve the first equation for \(t\):
\[ t=\frac{x-1}{2}. \]Substitute into the second equation:
\[ y= 5\left(\frac{x-1}{2}\right)-3. \]Therefore,
\[ \boxed{y=\frac52x-\frac{11}{2}}. \]Eliminating a Parameter from a Parabola
Consider
\[ x=t+2, \qquad y=t^2-1. \]From
\[ x=t+2, \]we get
\[ t=x-2. \]Substitute:
\[ y=(x-2)^2-1. \]Thus the Cartesian equation is
\[ \boxed{y=(x-2)^2-1}. \]Restrictions Can Be Lost During Elimination
Eliminating the parameter can reveal the Cartesian curve, but it may hide restrictions on which portion of the curve is actually traced.
Suppose
\[ x=t,\qquad y=t^2, \qquad t\ge0. \]Eliminating \(t\) gives
\[ y=x^2. \]But the parameter restriction implies
\[ x\ge0. \]Therefore only the right half of the parabola is traced.
Parametric Equations of a Line
A line through
\[ (x_0,y_0) \]with direction vector
\[ (a,b) \]can be written
\[ \boxed{ x=x_0+at,\qquad y=y_0+bt }. \]As \(t\) varies over all real numbers, the point moves along the entire line.
Example: Parametric Line
A line passes through
\[ (2,-1) \]with direction vector
\[ (3,4). \]Its parametric equations are
\[ \boxed{x=2+3t,\qquad y=-1+4t}. \]The corresponding slope is
\[ \frac{4}{3}. \]Recovering the Line Equation
From
\[ x=2+3t, \]solve for \(t\):
\[ t=\frac{x-2}{3}. \]Substitute into
\[ y=-1+4t: \] \[ y=-1+\frac43(x-2). \]Therefore,
\[ \boxed{y+1=\frac43(x-2)}. \]Vertical Lines Work Naturally in Parametric Form
Parametric equations do not require slope to exist.
The vertical line
\[ x=5 \]can be written
\[ \boxed{x=5,\qquad y=t}. \]As \(t\) varies, the \(x\)-coordinate remains fixed while the \(y\)-coordinate moves through all real values.
Line Segments with Restricted Parameters
A restricted parameter can trace only part of a line.
Suppose
\[ P=(x_1,y_1) \]and
\[ Q=(x_2,y_2). \]The segment from \(P\) to \(Q\) can be described by
\[ \boxed{ x=x_1+t(x_2-x_1), \qquad y=y_1+t(y_2-y_1), \qquad 0\le t\le1 }. \]At \(t=0\), the point is \(P\). At \(t=1\), it is \(Q\).
The Midpoint Appears at \(t=\frac12\)
In the segment parameterization,
\[ x=x_1+t(x_2-x_1), \qquad y=y_1+t(y_2-y_1), \]set
\[ t=\frac12. \]Then
\[ x=\frac{x_1+x_2}{2}, \qquad y=\frac{y_1+y_2}{2}. \]This recovers the midpoint formula naturally.
Parametric Circle
A circle of radius \(r\) centered at the origin can be parameterized by
\[ \boxed{ x=r\cos t,\qquad y=r\sin t }. \]The parameter \(t\) can be interpreted as an angle measured from the positive \(x\)-axis.
Why the Circle Parameterization Works
Substitute
\[ x=r\cos t \]and
\[ y=r\sin t \]into
\[ x^2+y^2. \]Then
\[ x^2+y^2 = r^2\cos^2 t+r^2\sin^2 t. \]Factor:
\[ r^2(\cos^2 t+\sin^2 t)=r^2. \]Therefore every generated point satisfies
\[ \boxed{x^2+y^2=r^2}. \]Orientation of the Standard Circle Parameterization
For
\[ x=r\cos t,\qquad y=r\sin t, \]the point begins at
\[ (r,0) \]when
\[ t=0. \]As \(t\) increases from \(0\) to \(2\pi\), the circle is traced once in the counterclockwise direction.
Translated Circle
A circle centered at
\[ (h,k) \]with radius \(r\) can be parameterized as
\[ \boxed{ x=h+r\cos t, \qquad y=k+r\sin t }. \]The constants \(h\) and \(k\) translate the origin-centered circle without changing its radius.
Parametric Ellipse
An axis-aligned ellipse centered at the origin,
\[ \frac{x^2}{a^2}+\frac{y^2}{b^2}=1, \]has the parameterization
\[ \boxed{ x=a\cos t,\qquad y=b\sin t }. \]Substitution gives
\[ \cos^2 t+\sin^2 t=1. \]Translated Ellipse
An ellipse centered at
\[ (h,k) \]can be written
\[ \boxed{ x=h+a\cos t, \qquad y=k+b\sin t }. \]Again, the parameters \(a\) and \(b\) determine scale while \(h\) and \(k\) determine position.
Parametric Parabola
The parabola
\[ y=x^2 \]has the simple parameterization
\[ \boxed{x=t,\qquad y=t^2}. \]More generally,
\[ y=a(x-h)^2+k \]can be parameterized by
\[ \boxed{x=h+t,\qquad y=at^2+k}. \]A Standard Parametric Hyperbola
The hyperbola
\[ \frac{x^2}{a^2}-\frac{y^2}{b^2}=1 \]can be parameterized using hyperbolic functions:
\[ \boxed{ x=a\cosh t,\qquad y=b\sinh t }. \]This works because
\[ \cosh^2 t-\sinh^2 t=1. \]This is a useful preview of how familiar trigonometric patterns extend to hyperbolic geometry and analysis.
Motion in the Plane
When \(t\) represents time,
\[ x=x(t),\qquad y=y(t) \]describe the position of a moving object.
The ordered pair
\[ \boxed{\mathbf{r}(t)=(x(t),y(t))} \]is called a position vector or position function.
Example: Constant-Velocity Motion
Suppose
\[ x=2+5t, \qquad y=7-3t. \]At \(t=0\), the object begins at
\[ (2,7). \]Each unit increase in time changes the coordinates by
\[ \Delta x=5, \qquad \Delta y=-3. \]The object therefore moves along a straight line with direction vector
\[ \boxed{(5,-3)}. \]Velocity Preview
In calculus, a parametric position
\[ \mathbf{r}(t)=(x(t),y(t)) \]leads naturally to the velocity vector
\[ \boxed{ \mathbf{v}(t)= \left( \frac{dx}{dt}, \frac{dy}{dt} \right) }. \]The derivative of each coordinate records its instantaneous rate of change.
This is one reason parametric equations are central in mechanics and multivariable calculus.
Slope of a Parametric Curve
When
\[ \frac{dx}{dt}\ne0, \]the slope of a parametric curve can be found from
\[ \boxed{ \frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}} }. \]This formula will be developed more fully in calculus, but its structure is already familiar: vertical rate divided by horizontal rate.
Different Parameterizations of the Same Curve
A geometric curve can have many different parameterizations.
For the unit circle, both
\[ x=\cos t,\qquad y=\sin t \]and
\[ x=\cos(2t),\qquad y=\sin(2t) \]trace the same set of points.
The second parameterization moves around the circle twice as quickly with respect to \(t\).
A Curve Can Be Traced More Than Once
Parametric equations may revisit the same point or retrace an entire curve.
For
\[ x=\cos t,\qquad y=\sin t, \]the interval
\[ 0\le t\le2\pi \]traces the circle once.
The interval
\[ 0\le t\le4\pi \]traces it twice.
Self-Intersections
A parametric curve can pass through the same geometric point at different parameter values.
If
\[ (x(t_1),y(t_1))=(x(t_2),y(t_2)) \]for
\[ t_1\ne t_2, \]the curve may have a self-intersection or may be retracing part of itself.
Parameters and Domains
A complete parametric description should specify the allowed parameter values when they matter.
The equations
\[ x=\cos t,\qquad y=\sin t \]with
\[ 0\le t\le\pi \]trace only the upper semicircle.
Changing the parameter domain can change the geometric image even when the coordinate formulas stay the same.
Why Parametric Form Is More Flexible Than \(y=f(x)\)
A vertical line cannot be represented as a single-valued function
\[ y=f(x), \]but it can easily be parameterized.
A circle also fails the vertical-line test as a whole, yet
\[ x=r\cos t,\qquad y=r\sin t \]describes it naturally.
Parametric form therefore handles many geometric paths that ordinary function notation cannot represent globally.
Parametric Form and Vector Form
The line equations
\[ x=x_0+at, \qquad y=y_0+bt \]can be combined into
\[ \boxed{ \mathbf{r}(t) = \begin{pmatrix} x_0\\ y_0 \end{pmatrix} + t \begin{pmatrix} a\\ b \end{pmatrix} }. \]Parametric and vector descriptions are therefore closely connected.
Three-Dimensional Parametric Curves
Parametric equations extend naturally into three dimensions:
\[ \boxed{ x=f(t),\qquad y=g(t),\qquad z=h(t) }. \]As \(t\) changes, the point
\[ (x,y,z) \]traces a curve through space.
Example: A Helix
Consider
\[ x=\cos t, \qquad y=\sin t, \qquad z=t. \]The \(x\)- and \(y\)-coordinates move around the unit circle while the \(z\)-coordinate steadily increases.
The resulting three-dimensional curve is a helix.
Common Error: Treating the Parameter as a Coordinate
In
\[ x=f(t),\qquad y=g(t), \]the plotted point is
\[ (x,y), \]not
\[ (x,y,t). \]In a two-dimensional parameterization, \(t\) generates the coordinates but is not itself a plotted coordinate.
Common Error: Eliminating the Parameter but Ignoring Restrictions
From
\[ x=t^2,\qquad y=t, \qquad t\ge0, \]we obtain
\[ x=y^2. \]But because
\[ t=y\ge0, \]only the portion with
\[ y\ge0 \]is traced.
Common Error: Assuming the Parameterization Is Unique
A curve does not have one mandatory parameterization.
For example,
\[ x=t,\qquad y=2t+1 \]and
\[ x=3s,\qquad y=6s+1 \]trace the same line.
The parameter name and rate of traversal can change without changing the underlying geometric locus.
Common Error: Ignoring Orientation
Two parameterizations can have the same Cartesian equation while tracing the curve in different directions.
Therefore converting to Cartesian form may lose information.
When direction matters, retain the original parameterization and parameter interval.
Common Error: Confusing Parameter Range with Coordinate Range
If
\[ 0\le t\le1, \]that does not automatically mean
\[ 0\le x\le1 \]or
\[ 0\le y\le1. \]Coordinate ranges must be determined from the actual functions \(x(t)\) and \(y(t)\).
A Reliable Procedure for Analyzing Parametric Equations
- Identify the parameter and its allowed domain.
- Calculate several representative parameter values.
- Determine the corresponding ordered pairs.
- Observe the direction of travel as the parameter increases.
- Eliminate the parameter when a Cartesian equation is useful.
- Carry over any restrictions that remain after elimination.
- Check whether the curve is traced once, repeatedly, or only partially.
- Use geometric structure to verify the result.
Practice 1: Generate a Point
For
\[ x=3t-1, \qquad y=t^2+2, \]find the point when
\[ t=2. \]Practice 2: Eliminate the Parameter
Eliminate \(t\) from
\[ x=t+4, \qquad y=2t-1. \]Practice 3: Parabola
Eliminate the parameter from
\[ x=t, \qquad y=3t^2-2. \]Practice 4: Parametric Line
Write parametric equations for the line through
\[ (1,-3) \]with direction vector
\[ (4,5). \]Practice 5: Line Segment
Write a parameterization for the line segment from
\[ A=(2,1) \]to
\[ B=(8,7). \]Practice 6: Circle
Write parametric equations for the circle centered at
\[ (3,-2) \]with radius
\[ 5. \]Practice 7: Ellipse
Write a standard parameterization for
\[ \frac{x^2}{16}+\frac{y^2}{9}=1. \]Practice 8: Parameter Restriction
Describe the portion of the curve traced by
\[ x=\cos t, \qquad y=\sin t, \qquad 0\le t\le\pi. \]Practice 9: Orientation
Compare
\[ x=\cos t,\qquad y=\sin t \]with
\[ x=\cos t,\qquad y=-\sin t. \]How does the orientation around the unit circle differ as \(t\) increases from \(0\)?
Practice 10: Identify the Error
A student eliminates \(t\) from
\[ x=t^2,\qquad y=t,\qquad t\ge0 \]and concludes that the parameterization traces the entire parabola
\[ x=y^2. \]Explain the error.
Practice Answers
Practice 1
At \(t=2\),
\[ x=3(2)-1=5, \] \[ y=2^2+2=6. \]Therefore,
\[ \boxed{(5,6)}. \]Practice 2
From
\[ x=t+4, \]we have
\[ t=x-4. \]Substitute:
\[ y=2(x-4)-1. \]Therefore,
\[ \boxed{y=2x-9}. \]Practice 3
Since
\[ x=t, \]substitute \(t=x\):
\[ \boxed{y=3x^2-2}. \]Practice 4
\[ \boxed{x=1+4t,\qquad y=-3+5t}. \]Practice 5
The displacement from \(A\) to \(B\) is
\[ (8-2,7-1)=(6,6). \]Therefore,
\[ \boxed{ x=2+6t,\qquad y=1+6t,\qquad 0\le t\le1 }. \]Practice 6
\[ \boxed{ x=3+5\cos t,\qquad y=-2+5\sin t }. \]Practice 7
\[ \boxed{ x=4\cos t,\qquad y=3\sin t }. \]Practice 8
The parameter begins at
\[ (1,0) \]and moves counterclockwise to
\[ (-1,0). \]Therefore it traces the
\[ \boxed{\text{upper semicircle}}. \]Practice 9
The first parameterization traces the unit circle counterclockwise from \((1,0)\).
The second traces the same circle clockwise from \((1,0)\).
Practice 10
Eliminating \(t\) gives
\[ x=y^2, \]but the restriction
\[ t\ge0 \]implies
\[ y\ge0. \]Therefore only the upper half of the sideways parabola is traced.
What to Remember
A parametric curve is described by
\[ \boxed{x=f(t),\qquad y=g(t)}. \]- the parameter generates the coordinate pair;
- parametric equations can describe curves that are not functions \(y=f(x)\);
- orientation records the direction in which the curve is traced;
- parameter restrictions determine which part of a curve is generated;
- eliminating the parameter can reveal a Cartesian equation but may lose orientation or restrictions;
- lines, circles, ellipses, parabolas, and spatial curves all admit useful parameterizations.
One especially important circle parameterization is
\[ \boxed{x=r\cos t,\qquad y=r\sin t}. \]Where We Go Next
Parametric equations introduce a new way to describe position by allowing coordinates to depend on a shared parameter. Another coordinate system changes the description more fundamentally by replacing horizontal and vertical coordinates with distance and direction.
The next lesson, Polar Coordinates, describes points using a radius and an angle and connects naturally with circles, rotation, trigonometry, and parametric curves.