Exponential Relationships

While linear models have a constant rate of change, exponential models describe situations where a quantity changes at a rate proportional to its current size.

• Exponential Growth: As the input increases steadily, the output increases by progressively larger amounts (e.g., the spread of a virus, or population growth).
• Exponential Decay: As the input increases steadily, the output decreases by progressively smaller amounts (e.g., the value of a car depreciating, or a hot drink cooling down).

1. The Structure of an Exponential Model

The defining feature of an exponential model is that the independent variable (usually time, $t$, or $x$) is the power (the exponent). Models typically take one of two forms:

• Without 'e': Written as $y = a(b)^x$.
  • $a$ is the starting (initial) value.
  • $b$ is the multiplier (e.g., a 12% increase means $b = 1.12$; a 15% decrease means $b = 0.85$).
• With 'e': Written as $y = Ae^{kx}$.
  • '$e$' is a mathematical constant (approximately 2.718) built into scientific calculators.
  • $A$ is the starting (initial) value.
  • If the power $kx$ is positive, the model shows growth. If the power is negative (e.g., $-kx$), the model shows decay.

2. Evaluating an Exponential Formula

"Evaluating" simply means substituting a given value for the independent variable into the formula to predict an outcome. Because the power involves an unknown, the numbers in exponential growth models get very large, very quickly.

3. Exponential Models in Spreadsheets (Excel)

• When entering a formula into Excel that contains '$e$', you must use the EXP command, and the power must be placed inside brackets. For example, the formula $y = 400e^{0.5x}$ would be typed into Excel as = 400 * EXP(0.5*A2).
• Crucial Excel Rule: Excel cannot produce an exponential trendline on a graph if your dataset includes the exact coordinate (0,0). The exponential trendline option will be completely greyed out. You must delete the (0,0) value from your columns to get the option back.