Exponential Functions — Growth, Decay & Graphs

An exponential function grows or decays by the same factor each time $x$ increases by $1$, with the variable sitting in the exponent rather than the base.

Exponential vs. polynomial, side by side

The key test is where the variable lives. In an exponential function it is in the exponent; in a polynomial it is the base. That one difference changes the whole growth pattern.

For real-valued exponential functions the usual conditions are $b > 0$ and $b \ne 1$. In the standard form

$a$ is the starting value and $b$ is the constant factor applied each time $x$ increases by $1$. So $f(x) = 3 \cdot 2^x$ is exponential, but $f(x) = 3x^2$ is not, because there the variable is the base.

Growth vs. decay: which base does what

For example, $2^x$ grows because each step multiplies by $2$, while $\left(\frac{1}{2}\right)^x$ decays because each step multiplies by $\frac{1}{2}$. The label depends on the base, not on a vague sense that the graph "eventually goes up."

How to read the graph quickly

The graph of a basic exponential function is smooth, not a set of disconnected points. A few features help you read it before computing many points:

1. It crosses $x = 0$ at $f(0) = a$, because $b^0 = 1$.
2. For the basic form with $a > 0$, the graph stays above the $x$-axis.
3. The line $y = 0$ is a horizontal asymptote, so the graph approaches the $x$-axis without touching it.
4. Growth graphs rise to the right; decay graphs fall to the right.

Worked example: graph $f(x) = 3 \cdot 2^x$

This example shows the starting value and the growth factor at once.

Find a few values:

Now the graph reads cleanly: the $y$-intercept is $(0, 3)$ so the initial value is $3$, each step to the right doubles the output because the base is $2$, and the graph still approaches $y = 0$ on the far left. Switch the base from $2$ to $\frac{1}{2}$ and the same setup turns into decay instead of growth.

When to use which model

Use an exponential model when change happens by a constant factor over equal intervals: compound interest, population growth under a fixed growth rate, radioactive decay, and cooling models. If the change is additive instead of multiplicative, an exponential model is usually not the right one.

Confusion points students hit

• Exponential vs. polynomial. $x^3$ is not exponential (variable is the base); $3^x$ is (variable is the exponent).
• Forgetting the base sets growth or decay. With $a > 0$, growth means $b > 1$ and decay means $0 < b < 1$.
• Forgetting the starting value. In $f(x) = a b^x$, the value at $x = 0$ is $a$.
• Factor vs. percent change. Growth of $20\%$ per step means a multiplier of $1.2$, not $0.2$; decay of $20\%$ means $0.8$.

To feel the contrast, take $f(x) = 5(0.7)^x$ and compute $f(0) = 5$, $f(1) = 3.5$, and $f(2) = 2.45$. The outputs shrink by the factor $0.7$ each step, which is exactly the difference between growth and decay.