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linear v. polynomial v. exponential v. logarithmic growth

Mathematical functions can be used to describe the change in a quantity over time. In the equations below, y represents the size of a quantity after x time has elapsed. b is some fixed number.

  • Linear growth occurs when there is a constant rate of change. The equation for a linear relationship is y = bx, and its graph is a straight line. If you work for an hourly wage, and don’t spend your earnings, your savings will grow linearly.
  • When the rate of change increases with time, numbers can grow more quickly. Quadratic and cubic growth can be represented by y = x2 and y = x3, respectively. The general form of this type of relationship can be written y = xb, and is called polynomial growth. The distance traveled by a falling object can be calculated with a quadratic equation.
  • Exponential (or geometric) growth is faster still. Here, the rate of growth is proportional to the value of y at any time. Exponential relationships can be expressed as y = bx. Bacterial populations with unlimited food, nuclear chain reactions, and computer processing power all grow exponentially.
  • Logarithmic growth is the inverse of exponential growth. Logarithmic phenomena grow very slowly, and have an equation of the form y = logbx. Sound volume and frequency are both perceived logarithmically, allowing humans to detect a huge range of sound levels.

Linear (red), cubic (blue), and exponential growth (green)

Exponential (green) and logarithmic growth (purple)

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