> ## Documentation Index > Fetch the complete documentation index at: https://docs.iotools.cloud/llms.txt > Use this file to discover all available pages before exploring further. # Parabola Calculator > Analyze y = ax² + bx + c: vertex, axis of symmetry, opening direction, x/y-intercepts, focus, and directrix, with a plotted diagram and step-by-step math. [Try Parabola Calculator in your browser →](https://iotools.cloud/tool/parabola-calculator/) ## OpenAPI ````yaml https://api.iotools.cloud/v1/openapi post /v1/tool/parabola-calculator openapi: 3.1.0 info: title: iotools.cloud API version: 1.0.0 description: >- Run any iotools.cloud tool over HTTP. Authenticate with `Authorization: Bearer iot_live_…`. **Only `POST /v1/tool/{slug}` costs credits.** Every GET here — the catalog, a tool's schema, your balance — is free. A tool call is charged its own weight or your plan's per-call minimum, whichever is larger; `x-iotools-credit-cost` on each operation is quoted at the free-tier minimum, and `GET /v1/tools/list` returns the exact figure for your key. `GET /v1/me/credits` reports what you have left, and `GET /v1/me/usage` reports what it went on. Failures are RFC 9457 problem documents — branch on `code`. servers: - url: https://api.iotools.cloud description: Production security: - bearerAuth: [] tags: - name: Catalog description: Find a tool and read its contract. Free. - name: Converters description: >- Convert between formats, encodings, and units — Base64, CSV and JSON, timestamps, and more. Fast, free, and processed right in your browser. - name: Formatters description: >- Format, beautify, minify, and validate code and data — JSON, HTML, CSS, SQL, and regex. Clean up messy input in one click, with nothing to upload. - name: Generators description: >- Generate exactly what you need and on demand — passwords, UUIDs, QR codes, hashes, random numbers, and more. Secure, instant, and free. - name: Calculators description: >- Crunch the numbers fast — from everyday math to specialized conversions and unit work. Free online calculators that run entirely in your browser. - name: Editors description: >- Edit and transform text, code, and images with quick, focused editors that run entirely in your browser — nothing to install, and no sign-up needed. - name: Utilities description: >- Everyday developer and web utilities — DNS and IP lookups, redirect and certificate checkers, and other quick diagnostics. Free and privacy-friendly. - name: Account description: Your key's allowance, limits and usage history. paths: /v1/tool/parabola-calculator: post: tags: - Calculators summary: Parabola Calculator description: >- Analyze y = ax² + bx + c: vertex, axis of symmetry, opening direction, x/y-intercepts, focus, and directrix, with a plotted diagram and step-by-step math. [Try Parabola Calculator in your browser →](https://iotools.cloud/tool/parabola-calculator/) operationId: run_parabola_calculator requestBody: required: true content: application/json: schema: type: object properties: coeffA: type: number description: a coeffB: type: number description: b coeffC: type: number description: c required: - coeffA - coeffB - coeffC examples: opens_upward_two_real_roots_y_x_4x_3: summary: Opens upward, two real roots (y = x² − 4x + 3) value: coeffA: '1' coeffB: '-4' coeffC: '3' repeated_root_vertex_on_the_x_axis_y_x_4x_4: summary: Repeated root — vertex on the x-axis (y = x² − 4x + 4) value: coeffA: '1' coeffB: '-4' coeffC: '4' opens_downward_no_real_roots_y_x_2x_5: summary: Opens downward, no real roots (y = −x² + 2x − 5) value: coeffA: '-1' coeffB: '2' coeffC: '-5' non_unit_leading_coefficient_y_2x_4x_6: summary: Non-unit leading coefficient (y = 2x² + 4x − 6) value: coeffA: '2' coeffB: '4' coeffC: '-6' responses: '200': description: Tool output content: application/json: examples: opens_upward_two_real_roots_y_x_4x_3: summary: Opens upward, two real roots (y = x² − 4x + 3) value: tool: parabola-calculator tool_version: 1.0.2 outputs: svgSource: >- VertexFocus summary: - property: Opens value: Upward (a > 0) - property: Vertex (h, k) value: (2, -1) - property: Axis of symmetry value: x = 2 - property: Y-intercept value: (0, 3) - property: X-intercepts value: (3, 0) and (1, 0) - property: Discriminant (Δ) value: '4' - property: Focus value: (2, -0.75) - property: Directrix value: y = -1.25 steps: |- y = 1x² - 4x + 3 (a = 1, b = -4, c = 3) Opens upward (a > 0) Vertex: h = −b / (2a) = 4 / 2 = 2 k = c − b² / (4a) = 3 − 16 / 4 = -1 Vertex (h, k) = (2, -1) Axis of symmetry: x = 2 Y-intercept: (0, 3) Discriminant: Δ = b² − 4ac = 4 Δ > 0 → two x-intercepts: x₁ = 3, x₂ = 1 Focus: (h, k + 1/(4a)) = (2, -1 + 1/4) = (2, -0.75) Directrix: y = k − 1/(4a) = -1 − 1/4 = -1.25 credits_used: 5 credits_remaining: null repeated_root_vertex_on_the_x_axis_y_x_4x_4: summary: Repeated root — vertex on the x-axis (y = x² − 4x + 4) value: tool: parabola-calculator tool_version: 1.0.2 outputs: svgSource: >- VertexFocus summary: - property: Opens value: Upward (a > 0) - property: Vertex (h, k) value: (2, 0) - property: Axis of symmetry value: x = 2 - property: Y-intercept value: (0, 4) - property: X-intercepts value: (2, 0) (double root) - property: Discriminant (Δ) value: '0' - property: Focus value: (2, 0.25) - property: Directrix value: y = -0.25 steps: |- y = 1x² - 4x + 4 (a = 1, b = -4, c = 4) Opens upward (a > 0) Vertex: h = −b / (2a) = 4 / 2 = 2 k = c − b² / (4a) = 4 − 16 / 4 = 0 Vertex (h, k) = (2, 0) Axis of symmetry: x = 2 Y-intercept: (0, 4) Discriminant: Δ = b² − 4ac = 0 Δ = 0 → the curve touches the x-axis once: x = 2 Focus: (h, k + 1/(4a)) = (2, 0 + 1/4) = (2, 0.25) Directrix: y = k − 1/(4a) = 0 − 1/4 = -0.25 credits_used: 5 credits_remaining: null opens_downward_no_real_roots_y_x_2x_5: summary: Opens downward, no real roots (y = −x² + 2x − 5) value: tool: parabola-calculator tool_version: 1.0.2 outputs: svgSource: >- VertexFocus summary: - property: Opens value: Downward (a < 0) - property: Vertex (h, k) value: (1, -4) - property: Axis of symmetry value: x = 1 - property: Y-intercept value: (0, -5) - property: X-intercepts value: None (curve doesn't cross the x-axis) - property: Discriminant (Δ) value: '-16' - property: Focus value: (1, -4.25) - property: Directrix value: y = -3.75 steps: >- y = -1x² + 2x - 5 (a = -1, b = 2, c = -5) Opens downward (a < 0) Vertex: h = −b / (2a) = -2 / -2 = 1 k = c − b² / (4a) = -5 − 4 / -4 = -4 Vertex (h, k) = (1, -4) Axis of symmetry: x = 1 Y-intercept: (0, -5) Discriminant: Δ = b² − 4ac = -16 Δ < 0 → no real x-intercepts (complex roots x = 1 + 2i, x = 1 - 2i) Focus: (h, k + 1/(4a)) = (1, -4 + 1/-4) = (1, -4.25) Directrix: y = k − 1/(4a) = -4 − 1/-4 = -3.75 credits_used: 5 credits_remaining: null non_unit_leading_coefficient_y_2x_4x_6: summary: Non-unit leading coefficient (y = 2x² + 4x − 6) value: tool: parabola-calculator tool_version: 1.0.2 outputs: svgSource: >- VertexFocus summary: - property: Opens value: Upward (a > 0) - property: Vertex (h, k) value: (-1, -8) - property: Axis of symmetry value: x = -1 - property: Y-intercept value: (0, -6) - property: X-intercepts value: (1, 0) and (-3, 0) - property: Discriminant (Δ) value: '64' - property: Focus value: (-1, -7.875) - property: Directrix value: y = -8.125 steps: |- y = 2x² + 4x - 6 (a = 2, b = 4, c = -6) Opens upward (a > 0) Vertex: h = −b / (2a) = -4 / 4 = -1 k = c − b² / (4a) = -6 − 16 / 8 = -8 Vertex (h, k) = (-1, -8) Axis of symmetry: x = -1 Y-intercept: (0, -6) Discriminant: Δ = b² − 4ac = 64 Δ > 0 → two x-intercepts: x₁ = 1, x₂ = -3 Focus: (h, k + 1/(4a)) = (-1, -8 + 1/8) = (-1, -7.875) Directrix: y = k − 1/(4a) = -8 − 1/8 = -8.125 credits_used: 5 credits_remaining: null schema: type: object required: - tool - tool_version - outputs - credits_used - credits_remaining properties: outputs: type: object properties: svgSource: type: string description: Diagram (SVG source) summary: type: array description: Analysis items: type: object properties: property: type: string description: Property value: type: string description: Value x-iotools-columns: - property - value steps: type: string description: Step-by-Step Solution tool: type: string description: The tool's slug, echoing the {slug} in the request path. tool_version: type: string description: Output-contract version for this tool. credits_used: type: integer description: >- Credits this call consumed, after any settlement refund. 0 when metering is disabled. credits_remaining: type: - integer - 'null' description: >- Credits left in the current monthly allowance, or null when metering is disabled. request_id: type: string description: Correlation id, also sent as x-request-id. '400': $ref: '#/components/responses/ValidationError' '401': $ref: '#/components/responses/Unauthorized' '402': $ref: '#/components/responses/InsufficientCredits' '403': $ref: '#/components/responses/ToolNotAllowed' '404': $ref: '#/components/responses/ToolNotFound' '413': $ref: '#/components/responses/PayloadTooLarge' '422': $ref: '#/components/responses/UnprocessableInput' '429': $ref: '#/components/responses/RateLimited' '500': $ref: '#/components/responses/ToolFailed' '503': $ref: '#/components/responses/ToolDisabled' security: - bearerAuth: [] components: responses: ValidationError: description: Invalid request. content: application/problem+json: schema: type: object description: >- RFC 9457 problem document, served as application/problem+json. Branch on `code`; `title` is human prose and may be reworded without notice. Some failures add extension members — `fields` on validation errors, `retry_after` on 429s, `credits_used`/`credits_remaining` on billing-adjacent failures — documented on the responses that carry them. required: - type - title - status - code properties: type: type: string format: uri description: Stable documentation URI for this failure. examples: - https://docs.iotools.cloud/errors/validation_error title: type: string description: Short human-readable summary of the failure. examples: - Invalid request status: type: integer description: HTTP status code, matching the response's own status. code: type: string description: >- Stable machine-readable error code — branch on this, not `title`. enum: - validation_error detail: type: string description: Human explanation of this occurrence. request_id: type: string description: Correlation id, also sent as x-request-id. fields: type: - object - string additionalProperties: type: string description: >- What failed: a map of field name → message. Absent when the body itself is malformed; a single string when the failure isn't tied to one field. example: type: https://docs.iotools.cloud/errors/validation_error title: Invalid request status: 400 code: validation_error detail: One or more inputs are invalid — see `fields`. request_id: e4042b29-8f1e-4c7a-9b52-6f0d1a3c7e11 fields: inputString: Required Unauthorized: description: Missing or invalid API key. content: application/problem+json: schema: type: object description: >- RFC 9457 problem document, served as application/problem+json. Branch on `code`; `title` is human prose and may be reworded without notice. Some failures add extension members — `fields` on validation errors, `retry_after` on 429s, `credits_used`/`credits_remaining` on billing-adjacent failures — documented on the responses that carry them. required: - type - title - status - code properties: type: type: string format: uri description: Stable documentation URI for this failure. examples: - https://docs.iotools.cloud/errors/invalid_api_key title: type: string description: Short human-readable summary of the failure. examples: - Invalid API key status: type: integer description: HTTP status code, matching the response's own status. code: type: string description: >- Stable machine-readable error code — branch on this, not `title`. enum: - invalid_api_key detail: type: string description: Human explanation of this occurrence. request_id: type: string description: Correlation id, also sent as x-request-id. example: type: https://docs.iotools.cloud/errors/invalid_api_key title: Invalid API key status: 401 code: invalid_api_key detail: 'Provide ''Authorization: Bearer ''.' request_id: e4042b29-8f1e-4c7a-9b52-6f0d1a3c7e11 InsufficientCredits: description: Monthly credit allowance exhausted. content: application/problem+json: schema: type: object description: >- RFC 9457 problem document, served as application/problem+json. Branch on `code`; `title` is human prose and may be reworded without notice. Some failures add extension members — `fields` on validation errors, `retry_after` on 429s, `credits_used`/`credits_remaining` on billing-adjacent failures — documented on the responses that carry them. required: - type - title - status - code properties: type: type: string format: uri description: Stable documentation URI for this failure. examples: - https://docs.iotools.cloud/errors/insufficient_credits - >- https://docs.iotools.cloud/errors/vendor_api_tier_limit_exceeded title: type: string description: Short human-readable summary of the failure. examples: - Insufficient credits - Free-tier API limit reached for this tool status: type: integer description: HTTP status code, matching the response's own status. code: type: string description: >- Stable machine-readable error code — branch on this, not `title`. enum: - insufficient_credits - vendor_api_tier_limit_exceeded detail: type: string description: Human explanation of this occurrence. request_id: type: string description: Correlation id, also sent as x-request-id. credits_used: type: integer description: Always 0 — a refused call charges nothing. credits_remaining: type: integer description: Credits left in the allowance — fewer than this call costs. example: type: https://docs.iotools.cloud/errors/insufficient_credits title: Insufficient credits status: 402 code: insufficient_credits detail: This call costs 1 credit and 0 remain in this month's allowance. request_id: e4042b29-8f1e-4c7a-9b52-6f0d1a3c7e11 credits_used: 0 credits_remaining: 0 ToolNotAllowed: description: Tool exists but has no API surface. content: application/problem+json: schema: type: object description: >- RFC 9457 problem document, served as application/problem+json. Branch on `code`; `title` is human prose and may be reworded without notice. Some failures add extension members — `fields` on validation errors, `retry_after` on 429s, `credits_used`/`credits_remaining` on billing-adjacent failures — documented on the responses that carry them. required: - type - title - status - code properties: type: type: string format: uri description: Stable documentation URI for this failure. examples: - https://docs.iotools.cloud/errors/tool_not_allowed title: type: string description: Short human-readable summary of the failure. examples: - Tool not available over the API status: type: integer description: HTTP status code, matching the response's own status. code: type: string description: >- Stable machine-readable error code — branch on this, not `title`. enum: - tool_not_allowed detail: type: string description: Human explanation of this occurrence. request_id: type: string description: Correlation id, also sent as x-request-id. example: type: https://docs.iotools.cloud/errors/tool_not_allowed title: Tool not available over the API status: 403 code: tool_not_allowed detail: >- "Background Remover" is available on iotools.cloud but has no API endpoint. request_id: e4042b29-8f1e-4c7a-9b52-6f0d1a3c7e11 ToolNotFound: description: No such tool. content: application/problem+json: schema: type: object description: >- RFC 9457 problem document, served as application/problem+json. Branch on `code`; `title` is human prose and may be reworded without notice. Some failures add extension members — `fields` on validation errors, `retry_after` on 429s, `credits_used`/`credits_remaining` on billing-adjacent failures — documented on the responses that carry them. required: - type - title - status - code properties: type: type: string format: uri description: Stable documentation URI for this failure. examples: - https://docs.iotools.cloud/errors/tool_not_found title: type: string description: Short human-readable summary of the failure. examples: - Tool not found status: type: integer description: HTTP status code, matching the response's own status. code: type: string description: >- Stable machine-readable error code — branch on this, not `title`. enum: - tool_not_found detail: type: string description: Human explanation of this occurrence. request_id: type: string description: Correlation id, also sent as x-request-id. example: type: https://docs.iotools.cloud/errors/tool_not_found title: Tool not found status: 404 code: tool_not_found detail: No tool with that slug. See GET /v1/tools/list. request_id: e4042b29-8f1e-4c7a-9b52-6f0d1a3c7e11 PayloadTooLarge: description: Body too large. content: application/problem+json: schema: type: object description: >- RFC 9457 problem document, served as application/problem+json. Branch on `code`; `title` is human prose and may be reworded without notice. Some failures add extension members — `fields` on validation errors, `retry_after` on 429s, `credits_used`/`credits_remaining` on billing-adjacent failures — documented on the responses that carry them. required: - type - title - status - code properties: type: type: string format: uri description: Stable documentation URI for this failure. examples: - https://docs.iotools.cloud/errors/payload_too_large title: type: string description: Short human-readable summary of the failure. examples: - Payload too large status: type: integer description: HTTP status code, matching the response's own status. code: type: string description: >- Stable machine-readable error code — branch on this, not `title`. enum: - payload_too_large detail: type: string description: Human explanation of this occurrence. request_id: type: string description: Correlation id, also sent as x-request-id. example: type: https://docs.iotools.cloud/errors/payload_too_large title: Payload too large status: 413 code: payload_too_large detail: Request body exceeds this tool's size limit. request_id: e4042b29-8f1e-4c7a-9b52-6f0d1a3c7e11 UnprocessableInput: description: Well-formed request, but there is no result to return. content: application/problem+json: schema: type: object description: >- RFC 9457 problem document, served as application/problem+json. Branch on `code`; `title` is human prose and may be reworded without notice. Some failures add extension members — `fields` on validation errors, `retry_after` on 429s, `credits_used`/`credits_remaining` on billing-adjacent failures — documented on the responses that carry them. required: - type - title - status - code properties: type: type: string format: uri description: Stable documentation URI for this failure. examples: - https://docs.iotools.cloud/errors/unprocessable_input title: type: string description: Short human-readable summary of the failure. examples: - Input cannot be processed status: type: integer description: HTTP status code, matching the response's own status. code: type: string description: >- Stable machine-readable error code — branch on this, not `title`. enum: - unprocessable_input detail: type: string description: Human explanation of this occurrence. request_id: type: string description: Correlation id, also sent as x-request-id. credits_used: type: integer description: >- Always 0 on `unprocessable_input` — a failed run is refunded, floor included. credits_remaining: type: integer description: >- Credits left after the refund. Absent when metering is disabled. example: type: https://docs.iotools.cloud/errors/unprocessable_input title: Input cannot be processed status: 422 code: unprocessable_input detail: >- This video has no captions or subtitles, so there is no transcript to extract. request_id: e4042b29-8f1e-4c7a-9b52-6f0d1a3c7e11 credits_used: 0 RateLimited: description: Per-minute rate limit exceeded. headers: Retry-After: description: Seconds to wait before retrying (RFC 9110 delta-seconds). schema: type: integer content: application/problem+json: schema: type: object description: >- RFC 9457 problem document, served as application/problem+json. Branch on `code`; `title` is human prose and may be reworded without notice. Some failures add extension members — `fields` on validation errors, `retry_after` on 429s, `credits_used`/`credits_remaining` on billing-adjacent failures — documented on the responses that carry them. required: - type - title - status - code properties: type: type: string format: uri description: Stable documentation URI for this failure. examples: - https://docs.iotools.cloud/errors/rate_limited title: type: string description: Short human-readable summary of the failure. examples: - Rate limit exceeded status: type: integer description: HTTP status code, matching the response's own status. code: type: string description: >- Stable machine-readable error code — branch on this, not `title`. enum: - rate_limited detail: type: string description: Human explanation of this occurrence. request_id: type: string description: Correlation id, also sent as x-request-id. retry_after: type: integer description: >- Seconds until the window resets — the same value as the `Retry-After` header. example: type: https://docs.iotools.cloud/errors/rate_limited title: Rate limit exceeded status: 429 code: rate_limited detail: Too many requests. Retry in 30s. request_id: e4042b29-8f1e-4c7a-9b52-6f0d1a3c7e11 retry_after: 30 ToolFailed: description: Tool failed to run. content: application/problem+json: schema: type: object description: >- RFC 9457 problem document, served as application/problem+json. Branch on `code`; `title` is human prose and may be reworded without notice. Some failures add extension members — `fields` on validation errors, `retry_after` on 429s, `credits_used`/`credits_remaining` on billing-adjacent failures — documented on the responses that carry them. required: - type - title - status - code properties: type: type: string format: uri description: Stable documentation URI for this failure. examples: - https://docs.iotools.cloud/errors/processing_error - https://docs.iotools.cloud/errors/internal_error title: type: string description: Short human-readable summary of the failure. examples: - Tool failed to run - Internal error status: type: integer description: HTTP status code, matching the response's own status. code: type: string description: >- Stable machine-readable error code — branch on this, not `title`. enum: - processing_error - internal_error detail: type: string description: Human explanation of this occurrence. request_id: type: string description: Correlation id, also sent as x-request-id. credits_used: type: integer description: >- Always 0 on `processing_error` — a failed run is refunded, floor included. credits_remaining: type: integer description: >- Credits left after the refund. Absent when metering is disabled. example: type: https://docs.iotools.cloud/errors/processing_error title: Tool failed to run status: 500 code: processing_error detail: The tool failed to run. Please try again. request_id: e4042b29-8f1e-4c7a-9b52-6f0d1a3c7e11 credits_used: 0 ToolDisabled: description: Tool temporarily disabled. content: application/problem+json: schema: type: object description: >- RFC 9457 problem document, served as application/problem+json. Branch on `code`; `title` is human prose and may be reworded without notice. Some failures add extension members — `fields` on validation errors, `retry_after` on 429s, `credits_used`/`credits_remaining` on billing-adjacent failures — documented on the responses that carry them. required: - type - title - status - code properties: type: type: string format: uri description: Stable documentation URI for this failure. examples: - https://docs.iotools.cloud/errors/tool_disabled - https://docs.iotools.cloud/errors/api_unconfigured title: type: string description: Short human-readable summary of the failure. examples: - Tool temporarily disabled - API not configured status: type: integer description: HTTP status code, matching the response's own status. code: type: string description: >- Stable machine-readable error code — branch on this, not `title`. enum: - tool_disabled - api_unconfigured detail: type: string description: Human explanation of this occurrence. request_id: type: string description: Correlation id, also sent as x-request-id. example: type: https://docs.iotools.cloud/errors/tool_disabled title: Tool temporarily disabled status: 503 code: tool_disabled detail: This tool is temporarily unavailable. Try again shortly. request_id: e4042b29-8f1e-4c7a-9b52-6f0d1a3c7e11 securitySchemes: bearerAuth: type: http scheme: bearer bearerFormat: iot_live_… ````