{
  "id": "rate-of-change",
  "version": "9805d832c984",
  "status": "published",
  "name": "Rate of Change Calculator",
  "question": "What is the rate of change?",
  "summary": "Computes the average rate of change, Δy ÷ Δx, between two points or across a table of values, with the rate on each interval, whether the rate is constant, and a chart.",
  "category": "math",
  "subcategory": "algebra",
  "url": "https://www.acalculator.org/math/rate-of-change-calculator",
  "markdown": "https://www.acalculator.org/math/rate-of-change-calculator.md",
  "kind": "function",
  "method": "Average rate of change = (y_last − y_first) ÷ (x_last − x_first); on each interval, (yₙ − yₙ₋₁) ÷ (xₙ − xₙ₋₁).",
  "assumptions": [
    "The points are taken in the order typed; the overall rate runs from the first point to the last.",
    "Each value is read as the exact decimal typed, so the rates are exact fractions."
  ],
  "inputs": {
    "$schema": "https://json-schema.org/draft/2020-12/schema",
    "type": "object",
    "properties": {
      "points": {
        "title": "Points (x, y)",
        "description": "The input x and the output y of each point, in order: two points, or a table of values.",
        "type": "array",
        "items": {
          "type": "object",
          "properties": {
            "x": {
              "title": "x",
              "description": "The input value, such as a year or a time.",
              "type": "number"
            },
            "y": {
              "title": "y",
              "description": "The output value at that x.",
              "type": "number"
            }
          }
        }
      }
    }
  },
  "outputs": {
    "rate": {
      "label": "Average rate of change",
      "description": "(last y − first y) ÷ (last x − first x).",
      "format": "number"
    },
    "fraction": {
      "label": "As a fraction",
      "description": "The average rate of change as an exact fraction in lowest terms.",
      "format": "text"
    },
    "dy": {
      "label": "Change in y (Δy)",
      "description": "Last y − first y.",
      "format": "number"
    },
    "dx": {
      "label": "Change in x (Δx)",
      "description": "Last x − first x.",
      "format": "number"
    },
    "percent": {
      "label": "Percent change in y",
      "description": "Δy ÷ |first y| × 100, when the first y is not 0.",
      "format": "percent"
    },
    "constant": {
      "label": "Is the rate constant?",
      "description": "Whether every interval has the same rate, so the points lie on one straight line.",
      "format": "text"
    },
    "intervals": {
      "label": "Rate on each interval",
      "description": "The average rate of change between each pair of neighbouring points, as a fraction.",
      "format": "text"
    }
  },
  "defaultAnswer": {
    "inputs": {
      "points": [
        {
          "x": 2007,
          "y": 2.84
        },
        {
          "x": 2009,
          "y": 2.41
        }
      ]
    },
    "outputs": {
      "rate": -0.215,
      "fraction": "-43/200",
      "dy": -0.43,
      "dx": 2,
      "percent": -15.140845070422534,
      "constant": "Two points: one interval.",
      "intervals": "(2007, 2.84) to (2009, 2.41): -43/200"
    },
    "text": "The average rate of change is -0.215 (-0.43 ÷ 2)."
  },
  "examples": [
    {
      "given": {
        "points": [
          {
            "x": 2007,
            "y": 2.84
          },
          {
            "x": 2009,
            "y": 2.41
          }
        ]
      },
      "expect": {
        "rate": -0.215,
        "fraction": "-43/200",
        "percent": -15.140845070422536
      },
      "source": "OpenStax Algebra and Trigonometry 2e, section 3.3, Rates of Change and Behavior of Graphs, Example 1: ($2.41 − $2.84) ÷ (2009 − 2007) ≈ −$0.22 per year (https://openstax.org/books/algebra-and-trigonometry-2e/pages/3-3-rates-of-change-and-behavior-of-graphs, retrieved 2026-10-01); exact value −0.215 by hand in content.mdx"
    },
    {
      "given": {
        "points": [
          {
            "x": 0,
            "y": 0
          },
          {
            "x": 1,
            "y": 1
          },
          {
            "x": 2,
            "y": 4
          },
          {
            "x": 3,
            "y": 9
          }
        ]
      },
      "expect": {
        "rate": 3,
        "constant": "No: the rate changes from one interval to the next.",
        "intervals": "(0, 0) to (1, 1): 1; (1, 1) to (2, 4): 3; (2, 4) to (3, 9): 5"
      },
      "source": "OpenStax Algebra and Trigonometry 2e, section 3.3 (average rate of change Δy ÷ Δx); hand calculation in content.mdx: (9 − 0) ÷ (3 − 0) = 3; intervals 1, 3, 5"
    },
    {
      "given": {
        "points": [
          {
            "x": 1,
            "y": 5
          },
          {
            "x": 3,
            "y": 11
          },
          {
            "x": 4,
            "y": 14
          }
        ]
      },
      "expect": {
        "rate": 3,
        "dy": 9,
        "dx": 3,
        "percent": 180,
        "constant": "Yes: every interval has the same rate, so the points lie on one line."
      },
      "source": "OpenStax Algebra and Trigonometry 2e, section 3.3 (average rate of change); hand calculation in content.mdx: (14 − 5) ÷ (4 − 1) = 3; both intervals 3"
    }
  ],
  "sources": [
    "OpenStax, Algebra and Trigonometry 2e, section 3.3, Rates of Change and Behavior of Graphs (average rate of change = Δy ÷ Δx = (y₂ − y₁) ÷ (x₂ − x₁); Example 1, gasoline prices 2007 to 2009), CC BY 4.0, retrieved 2026-10-01. https://openstax.org/books/algebra-and-trigonometry-2e/pages/3-3-rates-of-change-and-behavior-of-graphs"
  ],
  "related": [
    "slope",
    "instantaneous-rate-of-change",
    "percentage-change"
  ],
  "changelog": []
}