Essential Exponential Functions

Why This Matters

Exponential functions are the mathematical engine behind some of the most dramatic changes you'll encounter—populations that double, investments that compound, and radioactive materials that decay. In Unit 2 of AP Precalculus, you're being tested on your ability to recognize when a situation calls for exponential modeling and how these functions behave differently from the polynomial and rational functions you studied in Unit 1. The key insight? Exponential functions change proportionally to their current value, which is fundamentally different from linear or polynomial growth.

Understanding exponential functions also sets you up for success with logarithms, their inverse functions, which appear throughout the course and on the AP exam. You'll need to connect base values to growth vs. decay, transformations to graph behavior, and algebraic manipulation to real-world applications. Don't just memorize formulas—know what each parameter controls and why the function behaves the way it does. That conceptual understanding is what separates a 3 from a 5.

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Growth vs. Decay: The Base Determines Everything

The single most important feature of an exponential function is its base. When the base exceeds 1, the function grows; when it's between 0 and 1, the function decays. Both behaviors share the same fundamental structure—only the base value differs.

Exponential Growth Function

• Base $a > 1$—the function $f(x) = a^x$ increases without bound as $x$ increases, modeling situations like population growth or viral spread
• Y-intercept at $(0, 1)$ since $a^0 = 1$ for any positive base; this anchor point helps you sketch any exponential graph quickly
• Rate of change is proportional to current value—this self-reinforcing property creates the characteristic "J-curve" that accelerates over time

Exponential Decay Function

• Base $0 < a < 1$—the function $f(x) = a^x$ decreases toward zero as $x$ increases, modeling radioactive decay, depreciation, or cooling
• Horizontal asymptote at $y = 0$—the function approaches but never reaches zero, meaning some quantity always remains
• Same y-intercept at $(0, 1)$ as growth functions; the difference appears in the direction the curve moves as $x$ changes

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The Natural Exponential Function: Why $e$ Is Special

The constant $e \approx 2.718$ appears throughout mathematics because it produces the most "natural" rate of continuous change. Functions with base $e$ have unique calculus properties that make them the default choice for modeling continuous processes.

Natural Exponential Function

• $f(x) = e^x$ uses Euler's number—this irrational constant ($e \approx 2.71828...$) emerges naturally in continuous growth scenarios
• Inverse relationship with $\ln(x)$—the natural logarithm "undoes" the exponential, so $\ln(e^x) = x$ and $e^{\ln x} = x$
• Always positive and increasing—like all exponential growth functions, $e^x > 0$ for all real $x$, with the same y-intercept at $(0, 1)$

Continuous Compound Interest

• Formula $A = Pe^{rt}$—represents the limiting case of compound interest when compounding occurs infinitely often
• Variables: $P$ = principal, $r$ = annual rate (as decimal), $t$ = time in years—the result $A$ is the final amount
• Derived from discrete compounding—as compounding frequency $n \to \infty$ in $P(1 + r/n)^{nt}$, the expression converges to $Pe^{rt}$

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Core Properties: What Makes Exponentials Unique

Exponential functions share structural properties that distinguish them from polynomials and rational functions. These properties—domain, range, asymptotes, and one-to-one behavior—are frequently tested.

Domain and Range

• Domain is all real numbers $(-\infty, \infty)$—you can raise a positive base to any exponent, positive, negative, or zero
• Range is positive real numbers $(0, \infty)$—exponential outputs are always positive; they never touch or cross the x-axis
• One-to-one function—each input produces a unique output, which guarantees that the inverse (logarithm) exists

Horizontal Asymptote Behavior

• The x-axis ($y = 0$) is always the horizontal asymptote—for growth, the function approaches 0 as $x \to -\infty$; for decay, as $x \to +\infty$
• The function is continuous and smooth—no breaks, holes, or sharp corners; this contrasts with rational functions that have vertical asymptotes
• Asymptotic behavior shifts with transformations—adding a constant $k$ shifts the asymptote to $y = k$

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Transformations: Shifting, Stretching, and Reflecting

Transformations follow the same rules as other function families, but with exponentials, they dramatically change the graph's position and asymptote location. The general form $f(x) = a \cdot b^{(x-h)} + k$ captures all standard transformations.

Vertical and Horizontal Shifts

• Adding $k$ shifts the graph vertically—the horizontal asymptote moves from $y = 0$ to $y = k$, which is critical for graph interpretation
• Subtracting $h$ from $x$ shifts horizontally—$f(x) = b^{(x-h)}$ moves the graph $h$ units right (opposite of the sign)
• The y-intercept changes with shifts—recalculate by substituting $x = 0$ into the transformed function

Stretches and Reflections

• Multiplying by $a$ stretches or compresses vertically—if $|a| > 1$, the graph stretches; if $0 < |a| < 1$, it compresses
• Negative $a$ reflects over the x-axis—the function $f(x) = -b^x$ flips the graph, making outputs negative
• Reflection over the y-axis uses $f(x) = b^{-x}$—this is equivalent to using base $1/b$, converting growth to decay or vice versa

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Solving Exponential Equations

When the variable is in the exponent, you need logarithms to bring it down. The key strategy is isolating the exponential expression first, then applying the appropriate logarithm.

Using Logarithms to Solve

• Apply logarithms to both sides—if $b^x = c$, then $x = \log_b(c)$ or equivalently $x = \frac{\ln c}{\ln b}$
• Isolate the exponential first—before taking logarithms, get the exponential term alone (e.g., solve $3 \cdot 2^x = 24$ by first dividing to get $2^x = 8$)
• Check for extraneous solutions—especially when the original equation involves logarithms, verify answers don't produce negative arguments

The Logarithm-Exponential Relationship

• Logarithms are inverse functions—$\log_a(a^x) = x$ and $a^{\log_a x} = x$; this relationship is the foundation for solving exponential equations
• Common bases: 10 and $e$—$\log$ typically means base 10 (common log), while $\ln$ means base $e$ (natural log)
• Change of base formula: $\log_b x = \frac{\ln x}{\ln b}$—essential for calculator work since most calculators only have $\ln$ and $\log_{10}$

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Real-World Modeling Applications

Exponential functions model phenomena where change is proportional to current quantity. Recognizing when to apply exponential models—and which parameters to use—is a core AP skill.

Population and Growth Models

• $P(t) = P_0 \cdot b^t$ or $P(t) = P_0 e^{kt}$—$P_0$ is initial population, $b$ is growth factor per time unit, and $k$ is continuous growth rate
• Doubling time—the time required for a quantity to double; for continuous growth, doubling time $= \frac{\ln 2}{k}$
• Sustainability analysis—exponential models reveal whether growth rates are realistic long-term or will lead to resource depletion

Decay Applications

• Half-life—the time for a quantity to reduce by half; for radioactive decay $A(t) = A_0 \cdot (0.5)^{t/h}$, where $h$ is the half-life
• Depreciation models—assets losing a fixed percentage of value annually follow $V(t) = V_0(1-r)^t$, where $r$ is the depreciation rate
• Cooling and heating—Newton's Law of Cooling uses exponential decay to model temperature changes toward ambient conditions

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Quick Reference Table

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Self-Check Questions

1. What is the key difference between $f(x) = 3^x$ and $f(x) = (1/3)^x$, and how does this difference appear on their graphs?

2. If an investment uses continuous compounding at 5% annual interest, which formula applies, and how does the result compare to monthly compounding?

3. Compare and contrast the horizontal asymptote of $f(x) = 2^x$ with that of $g(x) = 2^x + 5$. What transformation causes this change?

4. Given the equation $5^{2x-1} = 125$, can you solve without logarithms? If so, how? If $5^{2x-1} = 100$, what changes?

5. A population doubles every 8 years. Write an exponential model for this situation and explain how you would find the population after 20 years.