AP Precalculus: Exponential Functions

Master exponential growth, decay, graphing, and real-world applications

📈 Growth 📉 Decay 📊 Graphing 💰 Applications

📚 Understanding Exponential Functions

Exponential functions model phenomena where quantities grow or decay by a constant percentage over equal time intervals. From population growth to radioactive decay to compound interest, exponential functions are essential for AP Precalculus and countless real-world applications.

1 Exponential Function Form

An exponential function has the form $f(x) = ab^x$, where the variable $x$ appears in the exponent. The function exhibits multiplicative behavior — output values change by a constant ratio for equal changes in input.

General Exponential Form \[f(x) = ab^x\]

Understanding the Parameters

$a$ — Initial Value

The y-intercept (value when $x = 0$)
$f(0) = a \cdot b^0 = a$

$b$ — Base (Growth Factor)

Must satisfy $b > 0$ and $b \neq 1$
Determines growth vs. decay

If $b > 1$: Exponential growth — output increases as $x$ increases
If $0 < b < 1$: Exponential decay — output decreases as $x$ increases
If $b = 1$: The function becomes constant ($f(x) = a$) — not exponential
If $a < 0$: The graph is reflected over the x-axis

📌 Examples

$f(x) = 3(2)^x$: $a = 3$, $b = 2 > 1$ → Growth, y-intercept = 3

$f(x) = 100(0.5)^x$: $a = 100$, $b = 0.5 < 1$ → Decay, y-intercept=100

$f(x) = -2(3)^x$: $a = -2$, $b = 3$ → Reflected growth

2 Domain, Range, and Asymptote

All exponential functions $f(x) = ab^x$ (with $a \neq 0$) have the same domain but a range that depends on the sign of $a$. They all have a horizontal asymptote at $y = 0$.

Domain

All real numbers: $(-\infty, \infty)$
You can substitute any $x$ value

Range (if $a > 0$)

Positive values only: $(0, \infty)$
Output never reaches or crosses 0

Range (if $a < 0$)

Negative values only: $(-\infty, 0)$
Graph is reflected below x-axis

Horizontal Asymptote

$y = 0$ (the x-axis)
Graph approaches but never touches

💡 Key Insight

Unlike polynomials, exponential functions never have x-intercepts (when $a > 0$) because $b^x$ is always positive for any real $x$.

3 Graphing Exponential Functions

Exponential graphs have distinctive shapes depending on whether they represent growth or decay. Key features include the y-intercept, asymptote, and the direction of the curve.

📈 Exponential Growth ($b > 1$)

Example: $f(x) = 2^x$

• Rises steeply to the right
• Approaches 0 on the left
• Increasing function
• Growth rate accelerates

📉 Exponential Decay ($0 < b < 1$)

Example: $f(x) = (0.5)^x$

• Falls steeply from the left
• Approaches 0 on the right
• Decreasing function
• Decay rate slows over time

Key Points for Any Exponential $f(x) = ab^x$

Y-intercept: $(0, a)$ — always pass through this point
Another easy point: $(1, ab)$ — when $x = 1$, $f(1) = ab$
Asymptote: $y = 0$ — the graph never crosses the x-axis
End behavior: One end approaches asymptote, other goes to $\pm\infty$

📌 Matching Functions to Graphs

To identify an exponential function from a graph:

1. Check the y-intercept to find $a$

2. Check direction: rising right → growth ($b > 1$); falling right → decay ($0 < b < 1$)

3. Use another point to find $b$: if $(1, y_1)$ is on graph, then $b = \frac{y_1}{a}$

4 Linear vs. Exponential Functions

Linear and exponential functions model fundamentally different types of change. Understanding the distinction is crucial for modeling real-world situations correctly.

Property	Linear: $f(x) = mx + b$	Exponential: $f(x) = ab^x$
Type of Change	Additive (constant difference)	Multiplicative (constant ratio)
Equal Intervals	Output changes by same amount	Output changes by same factor
Rate	Constant slope $m$	Rate changes (proportional to value)
Graph Shape	Straight line	Curved (concave up or down)
Long-term Behavior	Grows/decreases at constant rate	Grows/decreases faster and faster

📌 Identifying from Data

Given points: $(0, 5), (1, 10), (2, 20), (3, 40)$

Check differences: $10-5=5$, $20-10=10$, $40-20=20$ — NOT constant

Check ratios: $\frac{10}{5}=2$, $\frac{20}{10}=2$, $\frac{40}{20}=2$ — CONSTANT!

Conclusion: This is exponential with $a = 5$ and $b = 2$: $f(x) = 5(2)^x$

⚠️ AP Exam Alert

The AP exam frequently tests whether you can determine if data is linear (constant differences) or exponential (constant ratios). Always check both!

5 Exponential Growth & Decay Models

Exponential models are used when a quantity grows or decays by a constant percentage over equal time intervals. The growth/decay rate $r$ connects to the base $b$.

📈 Growth Model

$f(t) = a(1 + r)^t$

$r > 0$ = growth rate (as decimal)
$b = 1 + r > 1$
Examples: population, investments

📉 Decay Model

$f(t) = a(1 - r)^t$

$r > 0$ = decay rate (as decimal)
$b = 1 - r < 1$
Examples: depreciation, radioactive decay

📌 Example: Growth

Problem: A population of 1000 grows at 5% per year. Write the model and find the population after 10 years.

Model: $P(t) = 1000(1 + 0.05)^t = 1000(1.05)^t$

After 10 years: $P(10) = 1000(1.05)^{10} ≈ 1628.89$

📌 Example: Decay

Problem: A car worth $25,000 depreciates at 15% per year. Find its value after 4 years.

Model: $V(t) = 25000(1 - 0.15)^t = 25000(0.85)^t$

After 4 years: $V(4) = 25000(0.85)^4 ≈ \$13,050.16$

6 Continuous Exponential Model

When growth or decay happens continuously (every instant, not at discrete intervals), we use the natural base $e ≈ 2.71828$. This model is essential for calculus and many real-world applications.

Continuous Exponential Model \[f(t) = ae^{kt}\]

$a$

Initial value (when $t = 0$)

$e ≈ 2.71828$

Euler's number (natural base)

$k > 0$

Continuous growth

$k < 0$

Continuous decay

📌 Example

Problem: Bacteria grow continuously with $P(t) = 500e^{0.03t}$, where $t$ is in hours.

Initial population: $P(0) = 500e^0 = 500$

After 5 hours: $P(5) = 500e^{0.15} ≈ 580.91$

Growth rate: $k = 0.03$ or 3% continuous growth per hour

💡 Converting Between Models

To convert $ab^x$ to $ae^{kx}$: use $k = \ln(b)$. This means $b = e^k$.

7 Real-World Applications

Exponential functions model many important real-world phenomena. Here are the most common applications you'll encounter on the AP exam.

💰 Compound Interest

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

P = principal, r = rate, n = compounds/year, t = years

🏦 Continuous Compound

$A = Pe^{rt}$

Interest compounded every instant

☢️ Half-Life

$N(t) = N_0(\frac{1}{2})^{t/h}$

h = half-life, time for half to decay

🦠 Population Growth

$P(t) = P_0 e^{kt}$

Continuous growth model

📌 Half-Life Example

Problem: Carbon-14 has a half-life of 5730 years. If a sample has 100g, how much remains after 10,000 years?

Model: $N(t) = 100 \cdot (\frac{1}{2})^{t/5730}$

After 10,000 years: $N(10000) = 100 \cdot (0.5)^{10000/5730} ≈ 29.8$ grams

8 Transformations of Exponential Functions

Exponential functions can be transformed just like other functions. The general form with transformations is $f(x) = ab^{(x-h)} + k$.

$h$ — Horizontal Shift

$h > 0$: shift right
$h < 0$: shift left

$k$ — Vertical Shift

$k > 0$: shift up
$k < 0$: shift down

$a$ — Vertical Stretch/Reflect

$|a| > 1$: stretch
$a < 0$: reflect x-axis

New Asymptote

$y = k$ (shifted from $y = 0$)

📌 Example

Describe the transformations: $f(x) = 3(2)^{x-1} + 4$

• Base function: $y = 2^x$

• Vertical stretch: by factor of 3

• Horizontal shift: right 1 unit

• Vertical shift: up 4 units

• New asymptote: $y = 4$

📋 Quick Reference: Key Formulas

General Exponential

$f(x) = ab^x$

Percent Growth