Polynomial and Rational Functions: Definitions, Properties, and Graphing

Polynomial Functions

Definition of Polynomial Functions

A polynomial function in one variable is a function of the form:

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

• Coefficients $a_n, a_{n-1}, \ldots, a_0$ are constants called the coefficients of the polynomial.

• Degree: The highest exponent $n$ is the degree of the polynomial.

• Leading Term: The term $a_n x^n$ is the leading term.

• Leading Coefficient: The coefficient $a_n$ is the leading coefficient.

• Constant Term: The term $a_0$ is the constant term.

• Standard Form: A polynomial is in standard form when terms are written in descending order of degree.

Example: For $f(x) = 3x - 1 - 2x^5$, the leading term is $-2x^5$, the degree is 5, and the constant term is $-1$.

Identifying Polynomial Functions

• Check if all exponents are non-negative integers and coefficients are real numbers.

• Functions involving roots, negative exponents, or variables in denominators are not polynomials.

Example: $g(x) = \sqrt{x} + 4$ is not a polynomial because $\sqrt{x}$ is not a non-negative integer power of $x$.

Properties of Polynomial Functions

• The graph of a polynomial function is continuous and smooth (no sharp corners or cusps).

• Polynomial functions are defined for all real numbers.

• The end behavior of a polynomial is determined by its leading term.

Power Functions and End Behavior

A power function of degree $n$ is $f(x) = a_n x^n$. The end behavior depends on $n$ and the sign of $a_n$:

End Behavior of $f(x) = a_n x^n$	$a_n > 0$	$a_n < 0$
$n$ is even	$f(x) \to \infty$ as $x \to \pm\infty$	$f(x) \to -\infty$ as $x \to \pm\infty$
$n$ is odd	$f(x) \to -\infty$ as $x \to -\infty$, $f(x) \to \infty$ as $x \to \infty$	$f(x) \to \infty$ as $x \to -\infty$, $f(x) \to -\infty$ as $x \to \infty$

Zeros and Multiplicity

• If $r$ is a real zero of a polynomial, $(x - r)$ is a factor.

• If $(x - r)^m$ is a factor, $r$ is a zero with multiplicity $m$.

• If $m$ is odd, the graph crosses the x-axis at $r$; if $m$ is even, the graph touches but does not cross.

Example: For $f(x) = 2(x + 3)^2(x + 2)^3(x - 5)$, the zeros are $-3$ (multiplicity 2), $-2$ (multiplicity 3), and $5$ (multiplicity 1).

Turning Points

• The graph of a polynomial of degree $n$ has at most $n - 1$ turning points (points where the graph changes from increasing to decreasing or vice versa).

Constructing Polynomial Functions

• Given zeros and degree, construct the polynomial by multiplying factors $(x - r)$ for each zero $r$.

• Adjust the leading coefficient to satisfy additional conditions (e.g., passing through a specific point).

Example: Zeros: $-2, 3, -1$; degree: 3. Polynomial: $f(x) = a(x + 2)(x - 3)(x + 1)$, where $a$ is determined by other conditions.

Graphing Polynomial Functions

• Determine end behavior from the leading term.

• Find real zeros and their multiplicities.

• Find the y-intercept and additional points if necessary.

• Determine the greatest number of turning points.

• Plot points and use end behavior to sketch the graph.

Example: For $f(x) = -2x^2(x + 2)(x - 3)^2$, the end behavior is determined by $-2x^5$ (odd degree, negative leading coefficient).

Rational Functions

Definition of Rational Functions

A rational function is a function of the form:

$f(x) = \frac{p(x)}{q(x)}$

• $p(x)$ and $q(x)$ are polynomial functions.

• The domain is all real numbers except where $q(x) = 0$.

Example: $f(x) = \frac{x^2}{x + 1}$ has domain $x \neq -1$.

Finding the Domain of Rational Functions

• Set the denominator $q(x)$ equal to zero and solve for $x$.

• Exclude these values from the domain.

Example: $f(x) = \frac{1}{x^2 - 2x - 8}$, set $x^2 - 2x - 8 = 0$ to find excluded values.

Properties of Rational Functions

• May have vertical asymptotes where $q(x) = 0$.

• May have horizontal or oblique asymptotes depending on the degrees of $p(x)$ and $q(x)$.

Example: $f(x) = \frac{x^2 + 3}{x^2 + 16}$ has no real zeros in the denominator, so the domain is all real numbers.

Graphing Rational Functions

• Identify vertical asymptotes by setting $q(x) = 0$.

• Identify horizontal asymptotes by comparing degrees of $p(x)$ and $q(x)$.

• Plot intercepts and additional points as needed.

Additional info: For more advanced graphing, consider holes (removable discontinuities) where factors cancel in numerator and denominator.