3.3 Power Functions and Polynomial Functions

Power functions and polynomial functions form the foundation for nearly everything else in this unit. Power functions are the simplest building blocks (single-term expressions like $kx^n$), while polynomials combine multiple power-like terms into more complex curves. Getting comfortable with their structure, behavior, and differences now will pay off when you move into graphing, factoring, and solving polynomial equations.

Power Functions

Characteristics of power functions

A power function has the form $f(x) = kx^n$, where $k$ is a nonzero constant and $n$ is a real number. The constant $k$ stretches or compresses the graph vertically (and reflects it across the x-axis when negative), while $n$ controls the overall shape of the curve.

Domain and Range:

• When $n$ is a positive integer, the domain is all real numbers.
• When $n$ is a negative integer, $x = 0$ must be excluded (you'd be dividing by zero).
• When $n$ is a non-integer (like $\frac{1}{2}$), the domain may be restricted further (e.g., $f(x) = x^{1/2}$ only accepts $x \geq 0$).

The range depends on both $n$ and the sign of $k$:

• Even integer $n$, $k > 0$: range is $[0, \infty)$
• Even integer $n$, $k < 0$: range is $(-\infty, 0]$
• Odd integer $n$: range is all real numbers $(-\infty, \infty)$

Symmetry and Intercepts:

• Even exponents produce graphs symmetric about the y-axis (even functions).
• Odd exponents produce graphs symmetric about the origin (odd functions).
• A power function has at most one x-intercept at $x = 0$ and a y-intercept at $f(0) = k \cdot 0^n = 0$ (when $n > 0$). Don't confuse the y-intercept with $k$ itself; for positive exponents, the graph always passes through the origin.

End behavior of power functions

End behavior describes what happens to $f(x)$ as $x$ heads toward $\infty$ or $-\infty$. For power functions, the exponent $n$ and the sign of $k$ together determine this.

When $n > 0$ (positive exponent):

The end behavior mirrors what you'll see with polynomials, since the function is just a single term.

When $n < 0$ (negative exponent):

These create rational-type curves (like $f(x) = \frac{k}{x}$). As $x \to \pm\infty$, $f(x) \to 0$. The interesting behavior happens near $x = 0$, where the function shoots toward $\pm\infty$ depending on the sign of $k$ and whether $n$ is odd or even.

Polynomial Functions

Components of polynomial functions

A polynomial function has the general form:

$f(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0$

Each $a_i$ is a constant coefficient, and $n$ must be a non-negative integer. That integer requirement is what separates polynomials from general power functions.

Key vocabulary:

• Leading term: $a_n x^n$, the term with the highest power of $x$
• Leading coefficient: $a_n$, the coefficient of that leading term
• Degree: the value of $n$, the highest exponent

Common polynomial types by degree:

• Degree 0: constant (e.g., $f(x) = 5$)
• Degree 1: linear (e.g., $f(x) = 3x + 2$)
• Degree 2: quadratic (e.g., $f(x) = x^2 - 4x + 1$)
• Degree 3: cubic (e.g., $f(x) = 2x^3 - x$)

End behavior of polynomials is determined entirely by the leading term, because for very large $|x|$, the highest-degree term dominates all the others. This means you can use the same logic as power functions:

• Even degree, positive leading coefficient: both ends rise ($f(x) \to \infty$)
• Even degree, negative leading coefficient: both ends fall ($f(x) \to -\infty$)
• Odd degree, positive leading coefficient: falls left, rises right
• Odd degree, negative leading coefficient: rises left, falls right

For example, $f(x) = -2x^4 + 7x^2 - 3$ has even degree and a negative leading coefficient, so both ends point downward.

Power functions vs polynomial functions

These two types overlap but aren't the same thing. Every power function $kx^n$ where $n$ is a non-negative integer is also a polynomial (a one-term polynomial, called a monomial). But the categories diverge in important ways.

Similarities:

• Both are continuous on their domains.
• End behavior is driven by the highest-degree term and its coefficient.

Differences:

Analysis of polynomial functions

A few tools come up repeatedly when you analyze polynomials:

• Roots (zeros): The x-values where $f(x) = 0$. These are the points where the graph crosses or touches the x-axis. A polynomial of degree $n$ has at most $n$ real roots.
• Factoring: Rewriting a polynomial as a product of simpler expressions (like $(x - 2)(x + 3)$) is one of the main strategies for finding roots. If you can factor it, you can read the zeros directly.
• Turning points: A polynomial of degree $n$ can have at most $n - 1$ turning points (local maxima or minima). This connects to the derivative, which you may encounter later: the derivative of a polynomial tells you the rate of change and helps locate those turning points precisely.