Exponential Functions — Growth, Decay & Graphs

Exponential functions model repeated multiplication. In the standard form $f(x) = a b^x$, the variable is in the exponent, $a$ is the starting value, and $b$ is the constant factor that gets applied each time $x$ increases by $1$.

If $b > 1$, the function shows growth. If $0 < b < 1$, it shows decay. That is the main idea most students need first.

$$f(x) = a b^x$$

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

Exponential function definition

The key test is simple: the input variable, usually $x$, must be in the exponent. That is what makes the relationship multiplicative instead of additive.

So $f(x) = 3 \cdot 2^x$ is exponential, but $f(x) = 3x^2$ is not. In $3x^2$, the variable is the base, not the exponent.

This changes the pattern completely. Polynomial functions grow by powers of $x$. Exponential functions grow or shrink by the same factor each time $x$ increases by $1$.

Growth vs decay in exponential functions

In

$$f(x) = a b^x$$

the base $b$ controls the behavior:

• If $b > 1$, each step to the right multiplies the output by a number greater than $1$, so the graph grows.
• If $0 < b < 1$, each step to the right multiplies the output by a fraction, so the graph decays.

For example, $2^x$ grows because each step multiplies by $2$. But $\left(\frac{1}{2}\right)^x$ decays because each step multiplies by $\frac{1}{2}$.

How an exponential graph behaves

The graph of a basic exponential function is smooth, not made of disconnected points. A few features are worth spotting early:

1. It crosses the line $x = 0$ 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 gets closer and closer to the $x$-axis without touching it.
4. Growth graphs rise to the right. Decay graphs fall to the right.

These features let you read the graph quickly before you calculate many points.

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

This example shows the two most important ideas at once: the starting value and the growth factor.

$$f(x) = 3 \cdot 2^x$$

Start by finding a few values:

$$\begin{array}{c|ccccc} x & -2 & -1 & 0 & 1 & 2 \\ \hline f(x) & \frac{3}{4} & \frac{3}{2} & 3 & 6 & 12 \end{array}$$

Now the graph is easier to read:

• The y-intercept is $(0, 3)$, so the initial value is $3$.
• Each step to the right doubles the output, because the base is $2$.
• The graph rises faster and faster, but it still approaches $y = 0$ on the far left.

If you change the base from $2$ to $\frac{1}{2}$, the same setup turns into exponential decay instead of growth.

Common mistakes

Confusing exponential and polynomial functions

$x^3$ is not exponential. The variable is the base. In $3^x$, the variable is the exponent, so that is exponential.

Forgetting that the base sets growth or decay

In the standard form $a b^x$ with $a > 0$, growth means $b > 1$ and decay means $0 < b < 1$. The label depends on the base, not on a vague sense that the graph "eventually goes up."

Forgetting the starting value

In $f(x) = a b^x$, the value at $x = 0$ is $a$. That is the initial amount.

Mixing up the factor and the percent change

If a quantity grows by $20\%$ each step, the multiplier is $1.2$, not $0.2$. If it decays by $20\%$ each step, the multiplier is $0.8$.

When exponential functions are used

Exponential functions are used when change happens by a constant factor over equal intervals. Common examples include:

• compound interest
• population growth under a fixed growth rate
• radioactive decay
• cooling models and other decay processes

If the change is additive instead of multiplicative, an exponential model is usually not the right one.

Try a similar example yourself

Try your own version with $f(x) = 5(0.7)^x$. Compute $f(0)$, $f(1)$, and $f(2)$, then sketch the graph and check whether the outputs shrink by the same factor each step. That one change from base $2$ to base $0.7$ is enough to see the difference between growth and decay clearly.