Important Exponential Function Characteristics

Why This Matters

Exponential functions are the backbone of modeling real-world change—from population growth and radioactive decay to compound interest and viral spread. In AP Precalculus, you're being tested on your ability to analyze these functions structurally: understanding how the base determines behavior, why asymptotes exist, and how domain and range reflect the nature of exponential change. These concepts connect directly to transformations, inverse functions (logarithms), and limit behavior you'll encounter throughout the course.

Don't just memorize that $b > 1$ means growth—understand why the base controls everything, how the initial value anchors the function, and what the asymptote tells you about possible outputs. Exam questions will ask you to identify characteristics from graphs, write functions from scenarios, and compare exponential models. Know what concept each characteristic illustrates, and you'll be ready for both multiple choice and FRQs.

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The Foundation: Function Structure

Every exponential function follows the form $f(x) = a \cdot b^x$, where each component plays a specific role. The constant 'a' determines the initial value and vertical stretch, while the base 'b' controls the rate and direction of change.

Definition of an Exponential Function

• Standard form is $f(x) = a \cdot b^x$—'a' is the initial value (y-intercept when $x = 0$), and 'b' is the base that drives all behavior
• The exponent 'x' is the variable—this placement in the exponent (not the base) is what makes the function exponential rather than polynomial
• Models multiplicative change—each unit increase in x multiplies the output by factor b, unlike linear functions which add

Base (b) and Its Properties

• Base must satisfy $b > 0$ and $b \neq 1$—negative bases create undefined values for non-integer exponents; $b = 1$ yields constant function $f(x) = a$
• Base determines growth vs. decay—values of b directly control whether outputs increase or decrease as x increases
• Base affects steepness—larger bases (like 10) grow faster than smaller bases (like 2); bases closer to 1 change more gradually

Y-Intercept

• Always equals 'a' since $f(0) = a \cdot b^0 = a$—any base raised to zero power equals 1, leaving just the coefficient
• Represents initial value—in applications, this is your starting population, principal investment, or initial quantity
• Positive when $a > 0$, negative when $a < 0$—the sign of 'a' determines which side of the x-axis the function lives on

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Growth and Decay Behavior

The base determines whether your function increases or decreases, and how dramatically it changes. Understanding this distinction is essential for modeling and graph analysis.

Growth Factor ($b > 1$)

• When $b > 1$, the function increases—each unit increase in x multiplies the output by a factor greater than 1
• Larger bases mean steeper growth—$f(x) = 2^x$ doubles each step; $f(x) = 10^x$ increases tenfold
• Percent increase relates to base—a 5% growth rate means $b = 1.05$, since you keep 100% and add 5%

Decay Factor ($0 < b < 1$)

• When $0 < b < 1$, the function decreases—each unit increase in x multiplies by a fraction, shrinking the output
• Smaller bases (closer to 0) decay faster—$b = 0.1$ loses 90% per step; $b = 0.9$ loses only 10%
• Percent decrease relates to base—a 15% decay rate means $b = 0.85$, since you keep 85% of the previous value

End Behavior

• As $x \to \infty$ with $b > 1$: $f(x) \to \infty$—growth functions increase without bound toward positive infinity
• As $x \to -\infty$ for any valid base: $f(x) \to 0$—outputs approach but never reach the horizontal asymptote
• As $x \to \infty$ with $0 < b < 1$: $f(x) \to 0$—decay functions shrink toward the asymptote as x increases

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Domain, Range, and Asymptotes

These characteristics define where the function lives and what outputs are possible. They're directly connected to the algebraic structure of exponential functions.

Domain and Range

• Domain is all real numbers: $(-\infty, \infty)$—you can raise a positive base to any exponent, including negatives and fractions
• Range is $(0, \infty)$ when $a > 0$—positive bases raised to any power yield positive results only
• Range is $(-\infty, 0)$ when $a < 0$—a negative coefficient reflects the function below the x-axis

Horizontal Asymptote

• The line $y = 0$ (x-axis) is the horizontal asymptote—not vertical; this is a common exam trap
• Function approaches but never touches—as x moves toward the "flat" end, outputs get infinitely close to zero without reaching it
• Reflects impossibility of zero output—$b^x$ can never equal zero for any real x, so neither can $a \cdot b^x$

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Special Cases and Applications

These extend the basic exponential model to specific contexts you'll see on exams and in real-world problems.

Natural Exponential Function ($e^x$)

• Uses base $e \approx 2.71828$—an irrational constant that emerges naturally in continuous growth scenarios
• Derivative equals itself: $\frac{d}{dx}(e^x) = e^x$—this unique property makes it central to calculus
• Models continuous change—used in physics, biology, and finance when growth happens instantaneously rather than in steps

Compound Interest Application

• Formula: $A = P\left(1 + \frac{r}{n}\right)^{nt}$—P is principal, r is annual rate, n is compounding frequency, t is time in years
• Demonstrates exponential structure—the base is $\left(1 + \frac{r}{n}\right)$, the exponent is $nt$, and P is the initial value 'a'
• Continuous compounding uses $A = Pe^{rt}$—when n approaches infinity, the formula simplifies to the natural exponential form

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Quick Reference Table

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Self-Check Questions

1. If two exponential functions have the same y-intercept but different bases, how will their graphs compare at $x = 0$ versus $x = 5$?

2. A function has base $b = 0.75$. Without graphing, describe its end behavior as $x \to \infty$ and as $x \to -\infty$.

3. Compare and contrast: How does the horizontal asymptote of $f(x) = 3 \cdot 2^x$ relate to its range? Why can't the function ever output zero or negative values?

4. An investment grows according to $A = 5000(1.06)^t$. Identify the initial value, growth factor, and percent growth rate. What characteristic of exponential functions does the 5000 represent?

5. Why does the natural exponential function $f(x) = e^x$ appear so frequently in calculus and continuous growth models, rather than functions like $f(x) = 2^x$?