Use this original, deliberately narrow workbench: f(x) = 1/(x − a), with parameter a from −2 through 2 and a starting value of 1. Keep the plotted window fixed at x from −5 to 5 and y from −5 to 5. The reader's task is to explain the two branches and their gap, rather than enter an arbitrary expression and hope the graph is correct.
State the formula, parameter, and excluded input together
For the reciprocal family here, the denominator is zero at x = a. The full real domain is all real x except a. OpenStax explains denominator exclusions and the local behavior of reciprocal functions near a vertical asymptote. Rational functions
At the starting parameter a = 1, the formula is 1/(x − 1), and x = 1 is excluded. Put that statement beside the chart. A label saying only “x: −5 to 5” describes the viewing window; it does not describe the full domain. Values outside that window can still belong to the domain.
Build a five-row table from the positions a − 1, a − 0.5, a, a + 0.5, and a + 1:
| Input at a = 1 | Denominator x − a | Function value |
|---|---|---|
| 0 | −1 | −1 |
| 0.5 | −0.5 | −2 |
| 1 | 0 | Undefined |
| 1.5 | 0.5 | 2 |
| 2 | 1 | 1 |
Keep Undefined visible in its own row. Substituting zero would manufacture a point at (1, 0). An empty cell would leave the reader wondering whether the value was missing or the table failed.
These five points are exact checkpoints for this authored fixture. They are not a dense enough sample to establish every detail of the plotted curve or to prove its limiting behavior.
Split the branches deliberately
Specify one curve path for x less than a and another for x greater than a. Never connect a segment across a. This must remain true even if the plotting sample never lands exactly on the excluded input: skipping a division by zero is not sufficient when the next drawing command reconnects the two sides.
Add a visually distinct vertical guide at x = a and label it excluded-input asymptote. It is a guide, not part of the function's curve. If a horizontal y = 0 guide is shown, distinguish it too. For this fixed family, a numerator of one cannot give a function value of zero; approaching the axis does not mean reaching it. Reciprocal local and end behavior
Clip the curve to the declared viewing window without turning the clipped point into a mathematical endpoint. For example, at a = 1 and x = 1.1, the expected value is 10. It is defined but above the visible y limit. The screen edge does not change that value to 5, and it does not make x = 1.1 undefined.
Use three different explanations: defined and visible, defined but outside the viewing window, and undefined at the excluded input. The text and table should carry these distinctions even if the picture is small or unavailable.
Change the parameter without leaving an old answer behind
When a changes to −1, the excluded input moves to −1. The five table inputs become −2, −1.5, −1, −0.5, and 0. Their values remain −1, −2, Undefined, 2, and 1 because the offsets from a are unchanged.
This is also a useful stale-result check. At a = 1, a probe at x = 0 gives −1. At a = −1, the same probe gives 1. Update the formula label, excluded-input statement, table, branches, guide, and probe explanation from the current parameter. Do not leave a graph for the old parameter beside a table for the new one.
The example's probe accepts a finite x within −5 through 5. These probe bounds are an interface limit. Require a finite parameter within −2 through 2, and treat a blank, nonnumeric, nonfinite, or out-of-range field as invalid. Invalid input clears previous current output rather than silently defaulting to zero. A valid probe exactly equal to a reports Undefined and produces no point marker.
Reset restores a = 1, clears the probe input and actual observation fields, and rebuilds the default mathematical table and graph from that parameter. It should not populate a successful test report. This article specifies expected behavior; no plot, calculator, clipping algorithm, or browser interaction has been executed here.
Copyable reciprocal-graph preparation prompt
Prepare an original graph-workbench specification and mathematical acceptance ledger. Return readable text, not executable code or HTML. Do not claim a working plot or actual browser test.
Use only the fixed family f(x) = 1/(x-a). Accept finite a within -2 through 2, default 1. Declare the fixed viewing window x [-5,5] and y [-5,5]. State that the full real domain excludes x=a and is not limited to the displayed window.
Provide an exact point table at a-1, a-0.5, a, a+0.5, and a+1. The corresponding values are -1, -2, Undefined, 2, and 1. For a=1 show inputs 0, 0.5, 1, 1.5, 2. For a=-1 show inputs -2, -1.5, -1, -0.5, 0. Never replace Undefined with zero or an unexplained blank.
Specify two separate curve paths, one for x<a and one for x>a. No segment may cross a, even when sampling misses the exact excluded input. Specify a distinct vertical guide x=a labeled excluded-input asymptote, not part of the curve. An optional y=0 guide is not an achieved zero function value for this family.
Distinguish defined-and-visible, defined-but-offscreen, and undefined states. At a=1, x=1.1 has expected value 10, above the window rather than undefined or clamped to 5. Clipped edges are not roots or finite mathematical endpoints. A finite table is a checkpoint, not a proof of the entire curve.
Accept a finite probe x within -5 through 5. Blank, nonnumeric, nonfinite, or out-of-range input clears prior current output. A probe equal to a reports Undefined with no point marker. Parameter changes rebuild the formula, exclusion, branches, guides, table, and current valid probe together. Check x=0 gives -1 at a=1 and 1 at a=-1.
Reset restores a=1, clears probe input and actual observation fields, and rebuilds the default mathematical view. Leave runtime observations not tested. Provide a readable table/text equivalent and pending checks for clipping, branch separation, keyboard access, focus, and narrow layout. Do not add arbitrary expression parsing, eval, a generic solver, calculus proof, or financial and physical interpretations.
A separate way to view the prepared example
Keep the declared formula, exact point ledger, and branch conditions as the core explanation. NOS offers HTML saving and rendering as a separate place to view a prepared graph workbench. The actual graph still needs to be checked for false connecting segments, clipped values, and synchronized parameter changes.
Trust the picture only after you can explain which inputs are excluded and which values simply lie beyond its frame.