Exponential Relationships

An exponential relationship can be expressed in four equivalent forms. Being able to move between them fluently is the key skill in this topic — and NESA tests all four forms.

The four representations:

• Equation: $y = 3 \times 2^x$ — gives the exact rule; use it to substitute any $x$ value and find $y$.
• Table of values: list $(x, y)$ pairs: $(-1, 1.5),\ (0, 3),\ (1, 6),\ (2, 12),\ (3, 24)$. Notice each $y$-value is double the previous one — that's the growth factor $b = 2$.
• Set of ordered pairs: the same table written as $\{(-1, 1.5),\ (0, 3),\ (1, 6),\ (2, 12),\ (3, 24)\}$.
• Graph: plot the points and draw a smooth curve through them. The graph is curved (not straight), increasing, and has a horizontal asymptote at $y = 0$.

The shape of an exponential graph is determined entirely by $b$. Values of $b > 1$ give an increasing curve (growth); values $0 < b < 1$ give a decreasing curve (decay). The value of $A$ shifts the curve vertically, changing the $y$-intercept but not the shape.

Key features of $y = Ab^x$ ($b > 1$, growth):

• $y$-intercept at $(0,\ A)$ — always positive if $A > 0$
• Increasing at an increasing rate (the curve accelerates upward)
• Horizontal asymptote: $y = 0$ as $x \to -\infty$
• Domain: all real $x$; Range: $y > 0$ (if $A > 0$)

Key features of $y = Ab^{-x}$ (decay, or equivalently $y = A \left(\frac{1}{b}\right)^x$):

• $y$-intercept at $(0,\ A)$ — same starting point as the growth curve
• Decreasing at a decreasing rate (the curve flattens out)
• Horizontal asymptote: $y = 0$ as $x \to +\infty$
• Domain: all real $x$; Range: $y > 0$ (if $A > 0$)

An exponential model turns a real-world situation into an equation. Once you have the equation, you can predict values, interpret the starting condition, and critically evaluate where the model breaks down.

Constructing the model:

• Identify the initial value $A$ (the quantity at $x = 0$ — e.g. starting population, initial investment)
• Identify the growth or decay factor $b$ per period (e.g. doubles each hour → $b = 2$; reduces by 20% each year → $b = 0.8$ since $100\% - 20\% = 80\% = 0.8$)
• Write the equation: $y = Ab^x$ (growth) or $y = Ab^{-x}$ (decay)

Interpreting the $y$-intercept: Substituting $x = 0$ gives $y = A$. In context, $A$ is the value of the quantity at the start (time zero, beginning of the observation period). Always state $A$ with its units and meaning.

Limitations of exponential models:

• Growth models eventually become unrealistic — bacteria can't grow forever (food/space limits exist).
• Decay models approach zero but never reach it — in practice, quantities do reach zero.
• The model assumes a constant growth/decay factor — in reality, rates often change over time.
• Models should only be used within a reasonable domain — extrapolating too far gives nonsense predictions.