Exponential Functions: Graphs, Properties, and Applications

Exponential and Logarithmic Functions

Exponential Functions

Exponential functions are a fundamental class of functions in precalculus, characterized by a constant base raised to a variable exponent. They are widely used to model growth and decay in natural and applied sciences.

• Definition: An exponential function is of the form $f(x) = a^x$, where $a > 0$ and $a \neq 1$.

• Domain: All real numbers $(-\infty, \infty)$.

• Range: All positive real numbers $(0, \infty)$.

• Y-intercept: Always at $(0, 1)$ since $a^0 = 1$.

• X-intercept: None, as $a^x > 0$ for all real $x$.

• Horizontal Asymptote: The line $y = 0$ (x-axis).

• One-to-one: Each $x$ value corresponds to a unique $y$ value.

Graphing Exponential Functions

To graph an exponential function, evaluate the function at several values of $x$ and plot the corresponding points. The graph will show rapid growth or decay depending on the base $a$.

• For $a > 1$: The function increases as $x$ increases (exponential growth).

• For $0 < a < 1$: The function decreases as $x$ increases (exponential decay).

Properties of Exponential Functions

The properties of exponential functions depend on the value of the base $a$:

Property	$a > 1$	$0 < a < 1$
Domain	$(-\infty, \infty)$	$(-\infty, \infty)$
Range	$(0, \infty)$	$(0, \infty)$
Y-intercept	$(0, 1)$	$(0, 1)$
X-intercept	None	None
Horizontal Asymptote	$y = 0$	$y = 0$
Monotonicity	Increasing	Decreasing

Transformations of Exponential Functions

Exponential functions can be transformed by shifting, reflecting, or stretching/compressing their graphs. The general form is $f(x) = a^{bx + c} + d$.

• Horizontal Shifts: $f(x) = a^{x - h}$ shifts the graph $h$ units right.

• Vertical Shifts: $f(x) = a^x + k$ shifts the graph $k$ units up.

• Reflections: $f(x) = a^{-x}$ reflects the graph about the $y$-axis; $f(x) = -a^x$ reflects about the $x$-axis.

Exponential Growth and Decay

Exponential functions model many real-world phenomena, such as population growth, radioactive decay, and compound interest.

• Exponential Growth: $f(x) = Ca^x$, where $a > 1$ and $C$ is the initial value.

• Exponential Decay: $f(x) = Ca^x$, where $0 < a < 1$ and $C$ is the initial value.

The Number $e$

The number $e$ is a mathematical constant that is the base of the natural exponential function. It is an irrational number approximately equal to 2.71828.

• Definition: $e = \lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n$

• Properties: $e$ is used in continuous growth and decay models, such as continuously compounded interest.

Solving Exponential Equations

To solve exponential equations, express both sides with the same base if possible, then set the exponents equal to each other.

• Property: If $a^u = a^v$, then $u = v$.

• Example: Solve $9^{2x-5} = 27^x$.

Solution:

• Express both sides with base 3: $9 = 3^2$, $27 = 3^3$.

• Rewrite: $(3^2)^{2x-5} = (3^3)^x$

• Simplify: $3^{4x-10} = 3^{3x}$

• Set exponents equal: $4x-10 = 3x$

• Solve: $x = 10$

Example: Solving with Base $e$

Given $e^{-x^2} = e^{5x-6}$, set exponents equal:

• $-x^2 = 5x - 6$

• Rearrange: $x^2 + 5x - 6 = 0$

• Factor: $(x+6)(x-1) = 0$

• Solutions: $x = -6$ or $x = 1$

Summary Table: Properties of Exponential Functions

Base $a$	Monotonicity	Asymptote	Y-intercept
$a > 1$	Increasing	$y = 0$	$(0, 1)$
$0 < a < 1$	Decreasing	$y = 0$	$(0, 1)$