Graph hub
Plot functions and surfaces to sanity‑check results, find roots and intersections, and build intuition. When a graph matters, capture a snapshot and send your final expression into the Notebook.
Verification tip: Graphs are visual evidence, not proof. Use them to spot mistakes (wrong units, wrong mode, wrong domain),
then confirm with an analytic check or a second numerical method.
2D Graph
Functions · sliders · trace · roots · extrema · intersections
- Plot multiple expressions and compare behavior.
- Use trace to read values and pin points.
- Capture snapshots for reports and notebooks.
3D Graph
Surfaces · rotation · perspective · param sweeps
- Visualize z=f(x,y) surfaces and level sets.
- Rotate to understand gradients and curvature.
- Quickly detect discontinuities and clipping.
Quick recipes
Copy‑paste these into the 2D Graph tool (or adapt them). Use them as starting points for exploration.
Roots / zeros
y = x^2 - 4Look for x‑axis intersections (≈ ±2). Then verify by substitution.
Intersections
y = sin(x)y = 0.5Intersection points solve sin(x)=0.5. Confirm using an inverse trig check.
Parameter sweep (slider)
a = 0..10,0.1y = sin(a*x)Sweep a to see frequency changes. Check DEG/RAD if things look wrong.
Damping / stability intuition
y = exp(-0.25*x) * cos(6*x)Zoom out to see the envelope and confirm the decay rate qualitatively.
How to graph with confidence
A good graph answers a specific question: where is it zero, where does it cross another curve, how does it scale, and does it match a physical/units expectation?
- Start with a clean expression and a reasonable window (domain/range).
- Use trace + pinned points to read values precisely.
- Validate visually, then confirm with algebra or a second numerical method.
- Capture a snapshot and record assumptions in Notebook.
A repeatable workflow (recommended)
- Write the question first: root, intersection, extremum, asymptote, monotonicity, stability.
- Plot once with a wide window, then zoom until the feature you care about is clear.
- Use trace to extract candidate points, then verify by substitution or rearrangement.
Common mistakes (and how to catch them)
- Wrong angle mode: trig curves look "too fast" or "too slow" → check DEG vs RAD in Scientific.
- Bad scaling: you can hide behavior with an extreme window → reset and pick a tighter range.
- Domain issues: log/sqrt/reciprocal functions can explode or disappear → constrain x or avoid invalid regions.
- Over-trusting the picture: confirm key points numerically (substitution or solver).