VectorPaint reciprocal branches: check xy equals twelve without filling the zero gap

Connecting a reciprocal point list in one stroke can invent a line through the origin. Manually organise xy=12 in VectorPaint, checking products before visible ranges, open paths and labels. Distinguish asymptotes, clipping and true endpoints; a finite drawn segment is not the entire function.

Fix axes and the viewing window

Use an approximately 1700×1700 canvas, origin at screen (800,800), scale 80 px/unit on both axes and window -8..8 in each direction. Screen y is 800 minus 80 times mathematical y, not plus. Separate axes, samples, paths and notes, placing ticks first. Units are invented. The window limits viewing, not the domain; describe continuation beyond clipping.

Check each positive sample product

List positive pairs (2,6),(3,4),(4,3),(6,2), each product 12. Add window-edge pairs (1.5,8),(8,1.5). Place them in quadrant I, keeping pairs distinct. Sample (2,6) maps to screen (960,320). If points become collinear, recheck rather than changing them to fit a line. These six samples constrain an approximate drawing but do not prove every intermediate point satisfies the equation.

Keep both coordinates negative on the other branch

Negate both coordinates to obtain (-2,-6),(-3,-4),(-4,-3),(-6,-2), plus boundary pairs (-1.5,-8),(-8,-1.5). Products remain +12 in quadrant III. (-2,-6) maps to (640,1280). This is half-turn symmetry about the origin, not reflection in x; changing one sign gives -12. Negative coordinates are not negative physical lengths or material areas.

Use one open path per branch

Order each branch independently by increasing x and use separate open, unfilled paths through samples. Never bridge positive and negative groups or close/fill the gap. Check strokes before smoothing node tangents; control handles are not sample points. A labelled polyline approximation is valid without an invented function plotter. External values and manual placement define this exercise; smooth appearance proves no exact equation throughout.

Keep zero outside the sample set

From y=12/x, x≠0 and finite nonzero x also gives y≠0. Neither axis belongs to the curve. Approaching an axis is not joining it, and the window gives no origin point. Optional continuation arrows follow each branch end, not a bridge toward zero. Positive product excludes quadrants II/IV. This task fixes +12; a negative constant requires a different arrangement.

Label values and approximation limits

Label selected sample coordinates without crowding every point with arithmetic. Include xy=12 or y=12/x and x≠0. Mark the viewing window, not solid endpoints implying domain termination. Distinguish axes and paths by stroke/legend rather than merely faint colour. Keep a small product ledger checking both groups. Colours aid grouping but prove no sign condition. Retain sample coordinates separately from control handles for later corrections.

Troubleshoot signs, screen direction and bridging

For quadrant-IV placement check a single sign flip; for vertical inversion check screen-y subtraction. Split accidentally joined/closed paths rather than hiding an origin bridge with white. Reposition incorrect negative samples instead of reflecting across the wrong axis. Spot-check intermediate products and revise an inaccurate approximation, disclosing finite precision rather than deleting labels. After each change recheck samples and open-path state, retaining the native original.

Reopen and inspect both branches and the axis gap

Save native work, export PNG and reopen. Inspect separated branches and axes small, then signed samples, products and x≠0 enlarged. Repair an accidental filled region in path settings, not the function definition. Reopen independently selectable branches and editable samples/notes. Success explains product 12, quadrants and zero exclusion, distinguishing clipping from endpoints. It claims neither exact fitting, automatic asymptotes nor universal export validation; recalculate changed constants/scales.

Related tutorials and sources

VectorPaint manual line charts: plot coordinates before connecting one dataset; VectorPaint parabola diagrams: compare focus distance and perpendicular directrix distance.

Sources checked on 15 November 2026: the official VectorPaint App Store listing and Apple public lookup show version 1.0.8, requiring iOS / iPadOS 17.0 or later. Draw manually with published shapes, paths, nodes, layers and text. Conditions, scales and layouts are original exercise arrangements; check installed controls. Keep editable native work and PNG previews. SVG/PDF may retain any bitmap content used. External calculations and static diagrams are not new automatic proof, statistics or solver features. No completed device, physical-object or universal export testing is claimed.

Definition: OpenStax’s conic-sections chapter discusses an xy-type rectangular hyperbola. Twelve and sample coordinates are original; no textbook figure is copied or arbitrary Bézier segment claimed to equal a hyperbola exactly.