4.1 Exponential Functions

Exponential Functions

Exponential functions model situations where a quantity grows or shrinks by a constant multiplier over equal intervals. That's different from linear functions, which grow by a constant amount. This distinction matters because exponential behavior shows up everywhere: compound interest, population models, radioactive decay, and more.

more resources to help you study

practice questions

Exponential function fundamentals

An exponential function has the form $f(x) = b^x$, where $b$ is a positive constant called the base and $x$ is the exponent (the variable). The base determines the function's behavior:

• $b > 1$: exponential growth (the function increases as $x$ increases). Example: $f(x) = 2^x$ doubles every time $x$ increases by 1.
• $0 < b < 1$: exponential decay (the function decreases as $x$ increases). Example: $f(x) = (0.5)^x$ halves every time $x$ increases by 1.
• $b = 1$ just gives you $f(x) = 1$, a constant, so it's not interesting as an exponential function.
• $b$ cannot be negative or zero.

Properties of $f(x) = b^x$:

• Domain: all real numbers
• Range: all positive real numbers (the output is never zero or negative)
• Horizontal asymptote: $y = 0$ (the curve approaches the x-axis but never touches it)
• y-intercept: always $(0, 1)$, because $b^0 = 1$ for any valid base
• The function is always increasing when $b > 1$ and always decreasing when $0 < b < 1$

The natural exponential function $f(x) = e^x$ uses the special constant $e \approx 2.71828$. You'll see $e$ constantly in calculus and applied math. One reason it's special: the rate of change of $e^x$ at any point equals the value of $e^x$ at that point. For now in pre-calc, just know that $e$ is a specific irrational number (like $\pi$) that serves as the "natural" base for exponential and logarithmic work.

Exponential growth and decay equations

The general model for continuous exponential growth or decay is:

$A(t) = A_0 \cdot e^{kt}$

• $A_0$ = the initial amount (the value at $t = 0$)
• $k$ = the continuous growth/decay rate
• $t$ = time

If $k > 0$, the quantity grows. If $k < 0$, the quantity decays. Some textbooks write the decay version as $A(t) = A_0 e^{-kt}$ where $k$ is stated as a positive number and the negative sign is built into the formula.

Half-life is the time it takes for a decaying quantity to drop to half its current value:

$t_{1/2} = \frac{\ln(2)}{k}$

For example, carbon-14 has a decay constant of about $k = 0.000121$ per year, giving a half-life of roughly $\frac{0.693}{0.000121} \approx 5730$ years.

Doubling time is the growth equivalent. It's the time for a growing quantity to double:

$t_{\text{double}} = \frac{\ln(2)}{k}$

Notice the formula looks the same. The difference is context: half-life applies to decay situations, doubling time applies to growth situations.

A key idea to remember: in exponential functions, the rate of change is proportional to the current value. That's why growth accelerates over time and decay slows down over time.

Compound interest calculations

Compound interest is one of the most practical applications of exponential functions. Unlike simple interest (which only earns interest on the original principal), compound interest earns interest on previously accumulated interest too.

Periodic compounding formula:

$A = P\left(1 + \frac{r}{n}\right)^{nt}$

• $A$ = final amount (future value)
• $P$ = initial principal (present value, the amount you start with)
• $r$ = annual interest rate as a decimal (so 5% becomes 0.05)
• $n$ = number of compounding periods per year
• $t$ = time in years

Common values of $n$: annually = 1, quarterly = 4, monthly = 12, daily = 365.

Example: You invest $1,000 at 6% annual interest, compounded monthly, for 5 years.

$A = 1000\left(1 + \frac{0.06}{12}\right)^{12 \cdot 5} = 1000(1.005)^{60} \approx \$1348.85$

Continuous compounding formula:

$A = Pe^{rt}$

This is what happens when you let $n \to \infty$. Using the same example with continuous compounding:

$A = 1000 \cdot e^{0.06 \cdot 5} = 1000 \cdot e^{0.3} \approx \$1349.86$

The difference between monthly and continuous compounding is small here, but it grows with larger principals, higher rates, or longer time periods.

Exponential function graphs

To graph $f(x) = b^x$ by hand:

1. Plot the y-intercept at $(0, 1)$ (since $b^0 = 1$).
2. Calculate a few more points. For $f(x) = 2^x$, try $x = -2, -1, 1, 2, 3$ to get $(−2, 0.25)$, $(−1, 0.5)$, $(1, 2)$, $(2, 4)$, $(3, 8)$.
3. Draw the horizontal asymptote at $y = 0$ as a dashed line.
4. Connect the points with a smooth curve that approaches but never touches the asymptote.

Key features to recognize:

• y-intercept at $(0, 1)$ for the basic form $f(x) = b^x$
• Horizontal asymptote at $y = 0$
• The curve is steeper when the base is farther from 1. For instance, $3^x$ grows faster than $1.5^x$, and $(0.2)^x$ decays faster than $(0.8)^x$.
• The graph always passes through $(1, b)$ since $b^1 = b$.

Transformations of exponential functions:

• Vertical shift: $f(x) = b^x + k$ shifts the graph up by $k$ units (or down if $k < 0$). This also moves the horizontal asymptote to $y = k$.
• Horizontal shift: $f(x) = b^{x-h}$ shifts the graph right by $h$ units. The y-intercept moves, but the asymptote stays at $y = 0$.
• Reflection over the y-axis: $f(x) = b^{-x}$ flips the graph horizontally. Growth becomes decay and vice versa.
• Vertical stretch/compression: $f(x) = a \cdot b^x$ stretches the graph vertically by a factor of $a$ and changes the y-intercept to $(0, a)$.

When transformations are combined, the asymptote and intercepts shift accordingly. Always identify the new asymptote first, then plot a couple of transformed points to anchor your sketch.

Logarithms and inverse functions

Logarithms are the inverse of exponential functions. The statement $b^y = x$ is equivalent to $\log_b(x) = y$. In other words, a logarithm answers the question: "What exponent do you put on $b$ to get $x$?"

This inverse relationship is essential for solving exponential equations. If you need to solve $2^x = 32$, you can rewrite it as $x = \log_2(32) = 5$. For equations involving $e$, you'll use the natural logarithm $\ln$, which is just $\log_e$.

You'll explore logarithmic properties and equation-solving techniques in depth in the next sections of this unit. For now, the core idea is that exponentials and logarithms "undo" each other, just like squaring and square roots do.