From Parametric Equations to a Sketch
A parametrized curve describes points in the plane by two coordinate functions of a parameter. To sketch one, evaluate both functions at selected parameter values, plot the resulting ordered pairs, and connect the points in their parameter order. Arrows show the direction of traversal.
A table is a practical bridge between the equations and the drawing. It keeps the \(x\)- and \(y\)-coordinates paired correctly, makes the order of the points visible, and helps reveal features such as turning points or repeated points. The sketch is only as informative as the parameter values chosen: too few values can hide important changes in the curve.
A Practical Sketching Procedure
Begin with the parameter interval given in the problem. Choose values that cover it in order, including endpoints when they are included in the interval. Values spaced evenly are a useful starting point, but they are not always sufficient: add more near a sharp turn, a rapid change, or any feature that is not clear from the first plot.
Choose a manageable set in traversal order, paying attention to the ends of the interval.
For each parameter, calculate \(x=f(t)\) and \(y=g(t)\) separately, then record them as one ordered pair.
Use consistent axis scales and label the axes. Do not swap the horizontal and vertical coordinates.
Join neighboring points in parameter order with a smooth curve where the data supports one, then add arrows in the direction of increasing \(t\).
A table does not prove that the curve is made of straight segments between sampled points. It gives locations on the curve. Connecting neighboring points is a visual estimate of what happens between them; when the parameter values are far apart, that estimate may be poor.
Worked Example: A U-Shaped Trace
Worked Example: A U-Shaped Trace
Sketch the curve given by \(x=t\), \(y=t^2\), for \(-2\le t\le 2\). Use integer parameter values.
Evaluate the coordinates in increasing order of \(t\):
| \(t\) | \(x=t\) | \(y=t^2\) | Point \((x,y)\) |
|---|---|---|---|
| \(-2\) | \(-2\) | \((-2)^2=4\) | \((-2,4)\) |
| \(-1\) | \(-1\) | \((-1)^2=1\) | \((-1,1)\) |
| \(0\) | \(0\) | \(0^2=0\) | \((0,0)\) |
| \(1\) | \(1\) | \(1^2=1\) | \((1,1)\) |
| \(2\) | \(2\) | \(2^2=4\) | \((2,4)\) |
Plot the points from left to right in the listed order. They form a U-shaped sketch with its lowest sampled point at \((0,0)\). Place arrows from \((-2,4)\) toward \((-1,1)\), then toward \((0,0)\), and onward through \((1,1)\) to \((2,4)\). The arrows show the order of traversal, not a separate property of the plotted set.
Check: Substitution confirms the listed coordinates. For instance, at \(t=-2\), \(x=-2\) and \(y=(-2)^2=4\); at \(t=0\), \(x=0\) and \(y=0\); at \(t=2\), \(x=2\) and \(y=2^2=4\). Each table row uses the same parameter value for both coordinates.
Why the Parameter Order Matters
The curve is traced by allowing \(t\) to vary through its interval. Therefore, the order of the table rows records the order in which the parametrization visits its points. If the path turns back in the horizontal or vertical direction, sorting the plotted points by \(x\)- or \(y\)-coordinate can scramble that order.
Here is the basic reason connecting nearby parameter values is useful. When \(f\) and \(g\) are continuous, a small change in \(t\) produces small changes in both coordinates. Thus, for parameter values close together, the corresponding points are close together as well. A sequence of sufficiently close samples can therefore suggest the path between them. This justification relies on continuity; it does not say that the segment joining two samples is exactly part of the curve.
Worked Example: A Curve That Moves Right, Then Left
Worked Example: A Curve That Moves Right, Then Left
Sketch \(x=t^2-1\), \(y=t\), for \(-2\le t\le 2\), using integer values. Notice that \(x\) is not simply the parameter, so follow the table order rather than sorting by \(x\).
| \(t\) | \(x=t^2-1\) | \(y=t\) | Point \((x,y)\) |
|---|---|---|---|
| \(-2\) | \((-2)^2-1=4-1=3\) | \(-2\) | \((3,-2)\) |
| \(-1\) | \((-1)^2-1=1-1=0\) | \(-1\) | \((0,-1)\) |
| \(0\) | \(0^2-1=0-1=-1\) | \(0\) | \((-1,0)\) |
| \(1\) | \(1^2-1=1-1=0\) | \(1\) | \((0,1)\) |
| \(2\) | \(2^2-1=4-1=3\) | \(2\) | \((3,2)\) |
Plot the points in this order: \((3,-2)\), \((0,-1)\), \((-1,0)\), \((0,1)\), and \((3,2)\). The sketch moves left as the parameter increases from \(-2\) to \(0\), then moves right as it increases from \(0\) to \(2\). The vertical coordinate rises throughout, so the arrows point upward along the trace.
Check: At \(t=0\), the equations give \(x=0^2-1=-1\) and \(y=0\), so the middle point is \((-1,0)\). At \(t=2\), they give \(x=2^2-1=3\) and \(y=2\), so the final point is \((3,2)\). The other rows follow by the same direct substitutions shown in the table.
Sampling and Connecting Points Carefully
Choosing values evenly across the parameter interval often gives a useful first sketch. However, equal parameter steps do not necessarily produce equal distances between consecutive points. A curve may move slowly for one part of the interval and rapidly for another. If neighboring points appear far apart, use additional parameter values between them before deciding how the curve bends.
For trigonometric parametrizations, values at familiar angles often make exact coordinates possible. Such choices can reveal symmetry and repeated points without unnecessary decimal approximations. If exact coordinates include radicals, keep them exact while building the table; approximate only as needed for plotting, and use a consistent scale.
Also distinguish between a repeated point and a repeated row. Two different parameter values can produce the same coordinate pair. If that happens, the table should retain both parameter values because they record when the point is reached. The path may revisit a location, and a drawing that omits the order can conceal that fact.
Worked Example: A Circular Trace
Worked Example: A Circular Trace
Sketch \(x=2\cos t\), \(y=2\sin t\), for \(0\le t\le 2\pi\). Choose multiples of \(\frac{\pi}{4}\) so the coordinates can be evaluated exactly.
| \(t\) | \(x=2\cos t\) | \(y=2\sin t\) | Point \((x,y)\) |
|---|---|---|---|
| \(0\) | \(2(1)=2\) | \(2(0)=0\) | \((2,0)\) |
| \(\frac{\pi}{4}\) | \(2\left(\frac{\sqrt2}{2}\right)=\sqrt2\) | \(2\left(\frac{\sqrt2}{2}\right)=\sqrt2\) | \((\sqrt2,\sqrt2)\) |
| \(\frac{\pi}{2}\) | \(2(0)=0\) | \(2(1)=2\) | \((0,2)\) |
| \(\frac{3\pi}{4}\) | \(2\left(-\frac{\sqrt2}{2}\right)=-\sqrt2\) | \(2\left(\frac{\sqrt2}{2}\right)=\sqrt2\) | \((-\sqrt2,\sqrt2)\) |
| \(\pi\) | \(2(-1)=-2\) | \(2(0)=0\) | \((-2,0)\) |
| \(\frac{5\pi}{4}\) | \(2\left(-\frac{\sqrt2}{2}\right)=-\sqrt2\) | \(2\left(-\frac{\sqrt2}{2}\right)=-\sqrt2\) | \((-\sqrt2,-\sqrt2)\) |
| \(\frac{3\pi}{2}\) | \(2(0)=0\) | \(2(-1)=-2\) | \((0,-2)\) |
| \(\frac{7\pi}{4}\) | \(2\left(\frac{\sqrt2}{2}\right)=\sqrt2\) | \(2\left(-\frac{\sqrt2}{2}\right)=-\sqrt2\) | \((\sqrt2,-\sqrt2)\) |
| \(2\pi\) | \(2(1)=2\) | \(2(0)=0\) | \((2,0)\) |
Plot the first point at \((2,0)\), then plot each successive point around the origin in the table order. Connect them with a circular arc and add arrows moving from the rightmost point toward the top, then left, bottom, and back to the right. The endpoint matches the starting point, so the trace closes after the parameter reaches \(2\pi\).
Check: For \(t=\frac{\pi}{4}\), the equations give \(x=2\cos(\frac{\pi}{4})=\sqrt2\) and \(y=2\sin(\frac{\pi}{4})=\sqrt2\). For \(t=\frac{3\pi}{2}\), they give \(x=2\cos(\frac{3\pi}{2})=0\) and \(y=2\sin(\frac{3\pi}{2})=-2\). At both endpoints \(t=0\) and \(t=2\pi\), the point is \((2,0)\), as recorded.
A Common Mistake: Mixing Up Coordinates or Signs
A frequent error is to treat a coordinate function as though it were the parameter itself, or to make a sign error while evaluating a power. For example, for \(x=t^2-1\), substituting \(t=-2\) does not give \(x=-2-1=-3\). That incorrect calculation forgets that the formula squares \(t\).
Check each coordinate by returning to its own formula, and write the result as \((x,y)\), with the horizontal coordinate first. When plotting, use the same order: move \(x\) units horizontally and \(y\) units vertically. This simple check prevents a wrong point from redirecting the entire sketch.
What a Table Can and Cannot Tell You
A table gives exact locations at the selected parameter values and the order in which those locations occur. It can make a likely shape apparent, show whether the path revisits a point, and help identify where more samples would be useful. It cannot by itself guarantee that the connection between two widely separated points has been drawn correctly.
For a reliable sketch, make the plot reflect both the coordinate data and the parameter order. Use additional samples if the curve’s path between plotted points is unclear. Keep arrows consistent with the order in the table, and do not infer that the curve stops at an interior sample simply because the table has a row there.
Check Your Understanding
Use the coordinate functions and parameter intervals in each question. Focus on the table, the plot order, and the arrows.
- For \(x=t\), \(y=3-t\), with \(0\le t\le 3\), make a table at integer values of \(t\). What are the first and last plotted points?
- For \(x=t^2\), \(y=t\), with \(-2\le t\le 2\), calculate the points at \(t=-2\), \(t=0\), and \(t=2\). In what order should they be connected?
- Why can connecting two sampled points with a straight segment give an inaccurate picture of the curve between them?
- If the same point occurs for two different parameter values, should one of those parameter values be omitted from the table? Explain.
- For \(x=\cos t\), \(y=\sin t\), with \(0\le t\le 2\pi\), what point occurs at both endpoints, and what does that tell you about the sketch?