Does the description disclose side effects, auth requirements, rate limits, or destructive behavior?

No annotations are provided, so the description carries the full burden. It does disclose the deterministic outputs (equation, vertex, axis of symmetry, discriminant, asymptotes, monotonicity) and that missing params fall back to defaults, but it says nothing about side effects, error behavior for mismatched params/family, or whether results are purely computational and repeatable.

Agents need to know what a tool does to the world before calling it. Descriptions should go beyond structured annotations to explain consequences.

Is the description appropriately sized, front-loaded, and free of redundancy?

Three front-loaded sentences with no filler: purpose, parameter contract, return payload, and use case, in that order. The param examples are dense but every clause carries information; the sole mild cost is that the middle sentence is a long run-on mixing inputs and outputs.

Shorter descriptions cost fewer tokens and are easier for agents to parse. Every sentence should earn its place.

Given the tool's complexity, does the description cover enough for an agent to succeed on first attempt?

With no output schema, the description correctly compensates by enumerating the returned features, and it covers the family enum, param shape, defaults, and a use case. It stops short of describing failure modes (e.g. an unknown key in `params`) or confirming the operation is idempotent, but nothing essential for a correct call is missing.

Complex tools with many parameters or behaviors need more documentation. Simple tools need less. This dimension scales expectations accordingly.

Does the description clarify parameter syntax, constraints, interactions, or defaults beyond what the schema provides?

Schema coverage is already 100%, so baseline is 3, but the description genuinely adds meaning the schema does not: it maps which param keys belong to which family ({"a","b","c"} for quadratic, {"k"} for linear/inverse, {"a"} for exponential/logarithmic/power). That mapping would otherwise be guesswork, and the interaction between `family` and `params` is clarified. Only `locale` is left unaddressed, which the schema covers.

Input schemas describe structure but not intent. Descriptions should explain non-obvious parameter relationships and valid value ranges.

Does the description clearly state what the tool does and how it differs from similar tools?

States a specific verb ('plot and explore') and a specific resource ('a common function'), then enumerates the supported families, which cleanly separates it from graph-adjacent siblings like trig-lab, conic-sections, and geometry-lab. An agent can identify the tool's job without opening the schema.

Agents choose between tools based on descriptions. A clear purpose with a specific verb and resource helps agents select the right tool.

Does the description explain when to use this tool, when not to, or what alternatives exist?

Explicitly scopes usage: 'Use this for middle/high-school math when a user asks to draw or explain a function graph' — a clear trigger condition. It lacks any when-not guidance or named alternatives (e.g. trig-lab for trigonometric functions), so it falls short of a 5.

Agents often have multiple tools that could apply. Explicit usage guidance like "use X instead of Y when Z" prevents misuse.