Time Determines a Position
A parametric description can represent the position of a particle as time passes. At a particular time \(t\), the particle has a horizontal coordinate \(x(t)\) and a vertical coordinate \(y(t)\). Together, those values specify one point in the plane.
The parameter \(t\) tells us when the particle is at a position. The coordinates tell us where it is. A path records the locations reached, but by itself it does not record when each location is reached.
Keep the stated time interval in view. Even if formulas are defined for every real \(t\), a problem may ask about only a restricted interval. That restriction determines which positions belong to the path being considered.
Finding a Position from Coordinate Formulas
To find the location at a specified time, substitute that time into both coordinate functions. The result is one ordered pair: the \(x\)-coordinate first and the \(y\)-coordinate second. It is not a pair containing the time and one coordinate.
Worked Example: Evaluate a Position and Describe a Line Segment
A particle has position coordinates \(x(t)=2t-1\) and \(y(t)=t+2\) for \(0\leq t\leq 3\). Find its position at \(t=2\), and describe its path over the stated interval.
At \(t=2\), evaluate each coordinate:
Therefore, the position is \((3,4)\). To describe the path, first solve the \(x\)-equation for \(t\):
Substitute this expression into \(y=t+2\):
The time restriction \(0\leq t\leq 3\) gives the endpoint positions \((x(0),y(0))=(-1,2)\) and \((x(3),y(3))=(5,5)\). Thus the path is the portion of the line \(y=(x+5)/2\) from \((-1,2)\) to \((5,5)\), including both endpoints.
Check: At \(t=2\), the original formulas give \((3,4)\). This point satisfies the line equation because \((3+5)/2=4\), and it lies between the endpoint \(x\)-coordinates \(-1\) and \(5\).
The Path Depends on the Time Interval
Coordinate equations can describe a familiar curve, but the particle may trace only part of it. For example, sine and cosine coordinates satisfy a circle equation, yet a restricted interval may select only an arc. The equation of the curve does not by itself specify which portion is included.
Worked Example: Identify a Restricted Circular Path
Let \(x(t)=3\cos t\) and \(y(t)=3\sin t\), with \(0\leq t\leq \pi/2\). Find the positions at \(t=0\), \(t=\pi/4\), and \(t=\pi/2\), then describe the path.
At \(t=0\):
At \(t=\pi/4\), use \(\cos(\pi/4)=\sin(\pi/4)=\sqrt{2}/2\):
So the position is \(\left(3\sqrt{2}/2,3\sqrt{2}/2\right)\). At \(t=\pi/2\):
To identify the curve, square the coordinates and add:
Thus every position lies on the circle \(x^2+y^2=9\). On the specified interval, both \(\cos t\) and \(\sin t\) are nonnegative, so \(x\geq 0\) and \(y\geq 0\). The path is the quarter-circle in the first quadrant, from \((3,0)\) to \((0,3)\), including its endpoints.
Check: Each of the three listed positions satisfies \(x^2+y^2=9\). For the middle point, the check is \((3\sqrt{2}/2)^2+(3\sqrt{2}/2)^2=9/2+9/2=9\). The coordinate signs also agree with the first-quadrant restriction.
Reading Positions from a Table
A table gives the position at each listed time by placing the \(x\)- and \(y\)-values from the same row together. A sequence of rows shows sampled positions in time order. Unless the problem also specifies how the particle moves between those times, the table alone does not determine all intermediate positions or the complete path.
Worked Example: Use a Table and Its Coordinate Formulas
For \(0\leq t\leq 3\), a particle has \(x(t)=t^2-2t\) and \(y(t)=2t-2\). Find the position at \(t=2\), use the table to describe the sampled locations, and identify the curve containing the path.
Evaluate the coordinates at each integer time in the interval:
| Time \(t\) | \(x(t)\) | \(y(t)\) | Position \((x(t),y(t))\) |
|---|---|---|---|
| 0 | 0 | -2 | (0, -2) |
| 1 | -1 | 0 | (-1, 0) |
| 2 | 0 | 2 | (0, 2) |
| 3 | 3 | 4 | (3, 4) |
In particular, the position at \(t=2\) is \((0,2)\). To find a relation between the coordinates, note that \(y=2t-2\), so \(y/2=t-1\). Also, \(x=t^2-2t=(t-1)^2-1\). Therefore,
The path lies on this sideways parabola. The stated time interval gives \(-2\leq y\leq 4\), so the path is the portion corresponding to that \(y\)-range. For example, the \(t=2\) position lies on the curve because \(0=2^2/4-1\).
Check: Substituting \(t=2\) into the original formulas gives \((0,2)\), and substituting \(y=2\) into the coordinate relation gives \(x=2^2/4-1=0\). Both checks agree.
Why Coordinate Relations Describe the Path
The path is defined as the set of coordinate pairs produced by all allowed values of \(t\). A coordinate relation helps describe that set by expressing a condition that every such pair satisfies. The interval restriction must still be included: a relation can describe a larger curve than the actual path.
In the circle example, \(x^2+y^2=9\) describes the full circle, but the interval \(0\leq t\leq\pi/2\) restricts the positions to the first-quadrant arc. In the line example, the line equation alone extends infinitely in both directions, while the time interval selects a segment.
Reading a Simple Plot
On a coordinate plot, the horizontal and vertical axes represent \(x\) and \(y\), not time. Each point on the plotted path represents a location. If the graph marks parameter values or labels points with their times, those labels connect locations to times. For instance, a label \(t=2\) placed at \((3,4)\) means \(x(2)=3\) and \(y(2)=4\).
Read plot information carefully. A curve may cross itself, in which case one plotted location can correspond to more than one time. A plot without time labels may show the geometric path but not tell when the particle occupies each point. A handful of plotted points may likewise be samples rather than a complete description of the curve between them.
Identify the requested time or the full allowed interval.
At each time, use \(x(t)\) and \(y(t)\) together to form \((x(t),y(t))\).
Use the formulas, table, or plot to identify the points reached and any curve containing them.
State only the portion of the curve generated by allowed times.
Common Pitfall: Mixing Time with Coordinates
A frequent error is to treat \(t\) as one of the particle’s coordinates. In the first example, at \(t=2\), one might incorrectly write the position as \((2,3)\), using the time and the \(x\)-coordinate. But the second coordinate is \(y(2)\), not \(t\):
Another error is to report a full curve when the problem gives a restricted time interval. The circle equation \(x^2+y^2=9\) is correct for the second example, but the particle’s path over \(0\leq t\leq\pi/2\) is only the first-quadrant quarter-circle. State both the coordinate relation and the restriction that selects the actual path.
Why the Distinction Matters
Positions, paths, and time information answer different questions. A position answers where the particle is at one specified time. A path describes the collection of locations reached over an interval. The time-dependent formulas or labeled data show how those locations are associated with particular times. Keeping these ideas separate makes it easier to interpret parametric models correctly and to avoid conclusions that the given information does not support.
Check Your Understanding
For each question, distinguish the particle’s coordinates from its time information.
- If \(x(1)=-2\) and \(y(1)=5\), what is the particle’s position at \(t=1\)?
- For \(x(t)=t\) and \(y(t)=t^2\) on \(0\leq t\leq 2\), find the position at \(t=2\) and state a coordinate relation containing the path.
- Why does the equation of a full circle not necessarily give the entire path of a particle described by parametric equations?
- A table lists positions at \(t=0\) and \(t=1\). Does it, by itself, determine every position between those times? Explain.
- What information can a plot without time labels show, and what time-related information may be missing?
