Polynomial Functions and Their Graphs: College Algebra Study Guide

Polynomial Functions and Their Graphs

Definition of a Polynomial Function

A polynomial function is a function of the form $f(x) = a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0$, where $a_n \neq 0$ and $n$ is a non-negative integer. The coefficients $a_n, a_{n-1}, \ldots, a_0$ are real numbers, and $n$ is called the degree of the polynomial.

• Degree: The highest power of $x$ in the polynomial.

• Leading Coefficient: The coefficient of the term with the highest degree.

• Constant Term: The term without $x$ (i.e., $a_0$).

Graphs of Polynomial Functions – Smooth and Continuous

Polynomial functions of degree 2 or higher have graphs that are smooth (no sharp corners) and continuous (no breaks; can be drawn without lifting the pencil).

End Behavior of Polynomial Functions

The end behavior describes how the graph behaves as $x$ approaches positive or negative infinity. The sign of the leading coefficient and the degree of the polynomial determine this behavior.

• For large $|x|$, the highest-degree term dominates the graph's behavior.

• Odd-degree polynomials: Opposite behavior at each end.

• Even-degree polynomials: Same behavior at each end.

The Leading Coefficient Test

The Leading Coefficient Test helps predict the end behavior of a polynomial function based on its degree and leading coefficient.

• If $n$ is odd and $a_n > 0$: Falls left, rises right.

• If $n$ is odd and $a_n < 0$: Rises left, falls right.

• If $n$ is even and $a_n > 0$: Rises both ends.

• If $n$ is even and $a_n < 0$: Falls both ends.

The following table summarizes the Leading Coefficient Test:

Leading Term	Odd n	Even n
$a_n > 0$	Falls left, rises right	Rises both ends
$a_n < 0$	Rises left, falls right	Falls both ends

Example: Using the Leading Coefficient Test

Consider $f(x) = x^3 + 3x^2 - x - 3$. The degree is 3 (odd), and the leading coefficient is 1 (positive). Thus, the graph falls to the left and rises to the right.

For $f(x) = -4x^2(x-1)^2(x+5)$, the degree is $2+2+1=5$ (odd), and the leading coefficient is -4 (negative). Thus, the graph rises left and falls right.

Application Example: Modeling with Polynomial Functions

Polynomial functions can model real-world phenomena, but their end behavior may limit their applicability. For example, a model for AIDS cases in the U.S. uses a cubic polynomial:

• $f(x) = -49x^3 + 806x^2 + 3776x + 2503$

• Degree is 3 (odd), leading coefficient is -49 (negative).

• Graph falls right, rises left. Eventually, the model predicts negative cases, which is not realistic.

Zeros of Polynomial Functions

The zeros (or roots) of a polynomial function are the values of $x$ for which $f(x) = 0$. These correspond to the $x$-intercepts of the graph.

• To find zeros, set $f(x) = 0$ and solve for $x$.

• Each real zero appears as an $x$-intercept.

Example: Finding Zeros of a Polynomial Function

Find all zeros of $f(x) = x^4 - 4x^2$:

• Factor: $x^4 - 4x^2 = x^2(x^2 - 4) = x^2(x-2)(x+2)$

• Zeros: $x = 0, 2, -2$

Multiplicity and x-Intercepts

The multiplicity of a zero refers to how many times a particular root appears in the factorization of the polynomial.

• If a zero has even multiplicity, the graph touches the $x$-axis and turns around at that point.

• If a zero has odd multiplicity, the graph crosses the $x$-axis at that point.

• Graphs flatten out near zeros with multiplicity greater than one.

Example: Finding Zeros and Their Multiplicities

For $f(x) = (x+1)(2x-3)^2$, $x = -1$ has multiplicity 1 (crosses $x$-axis), $x = 1.5$ has multiplicity 2 (touches and turns).

The Intermediate Value Theorem

The Intermediate Value Theorem states: If $f$ is a polynomial function with real coefficients and $f(a)$ and $f(b)$ have opposite signs, then there is at least one value $c$ between $a$ and $b$ for which $f(c) = 0$.

• Guarantees at least one real root between $a$ and $b$ if $f(a)$ and $f(b)$ have opposite signs.

Example: Using the Intermediate Value Theorem

Show that $f(x) = x^3 - 2x - 5$ has a real zero between 2 and 3:

• $f(2) = 2^3 - 2 \cdot 2 - 5 = 8 - 4 - 5 = -1$ (negative)

• $f(3) = 3^3 - 2 \cdot 3 - 5 = 27 - 6 - 5 = 16$ (positive)

• Since $f(2)$ and $f(3)$ have opposite signs, there is a real zero between 2 and 3.

Turning Points of Polynomial Functions

A turning point is a point where the graph changes direction from increasing to decreasing or vice versa. For a polynomial of degree $n$, the graph has at most $n-1$ turning points.

• Maximum number of turning points: $n-1$

• Turning points correspond to local maxima and minima.

Graphing a Polynomial Function: Five-Step Strategy

To graph a polynomial function, follow these steps:

1. Use the Leading Coefficient Test to determine end behavior.

2. Find x-intercepts by solving $f(x) = 0$ and analyzing multiplicity.

3. Find the y-intercept by computing $f(0)$.

4. Use symmetry if applicable: $f(-x) = f(x)$ (y-axis symmetry), $f(-x) = -f(x)$ (origin symmetry).

5. Check turning points: Maximum is $n-1$ for degree $n$.

Example: Graph $f(x) = 2(x+2)^2(x-3)$

• Degree: 3 (odd), leading coefficient: 2 (positive)

• End behavior: Falls left, rises right

• x-intercepts: $x = -2$ (multiplicity 2, touches and turns), $x = 3$ (multiplicity 1, crosses)

• y-intercept: $f(0) = -24$

• No symmetry

• Maximum turning points: 2

Additional info: The five-step strategy ensures a complete and accurate graph, accounting for all critical features of polynomial functions.