Polynomial Functions and Their Graphs: Study Notes for College Algebra

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 $a_n, a_{n-1}, \ldots, a_0$ are real numbers and $n$ is a non-negative integer. The highest power of $x$ is called the degree of the polynomial.

• Degree: The largest exponent of $x$ in the function.

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

• Example: $f(x) = 2x^3 + 5x^2 - x + 7$ is a polynomial of degree 3 with leading coefficient 2.

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).

• Graphs contain only rounded curves.

• Graphs have no gaps or jumps.

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.

• End behavior is crucial for sketching polynomial graphs.

The Leading Coefficient Test

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

• Odd degree: Graphs have opposite behavior at each end.

• Even degree: Graphs have the same behavior at each end.

• Positive leading coefficient: Graph rises to the right.

• Negative leading coefficient: Graph falls to the right.

Summary Table:

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

Examples: Using the Leading Coefficient Test

To determine end behavior, identify the sign of the leading coefficient and the degree of the polynomial.

• Example: $f(x) = x^3 + 3x^2 - x - 3$ has degree 3 (odd) and leading coefficient 1 (positive). The graph falls to the left and rises to the right.

When the polynomial is in factored form, add the exponents to find the degree. The sign of the leading coefficient determines the direction.

• Example: $f(x) = -4x^2(x-1)^2(x+5)$ has degree $2+2+1=5$ (odd), leading coefficient -4 (negative). The graph rises left and falls right.

Application Example: Modeling with Polynomial Functions

Polynomial functions can be used to model real-world phenomena, such as population growth or disease cases. However, the end behavior may limit the model's applicability over time.

• Example: $f(x) = -49x^3 + 806x^2 + 3776x + 2503$ models AIDS cases in the U.S. The degree is 3 (odd), leading coefficient is -49 (negative), so the graph rises left and falls right.

• Model breakdown occurs when the graph predicts impossible values (e.g., negative cases).

Zeros of Polynomial Functions

The zeros of a polynomial function are the values of $x$ for which $f(x) = 0$. These are also called roots or solutions of the equation $f(x) = 0$.

• Each real root appears as an x-intercept on the graph.

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

Example: Finding Zeros

• For $f(x) = x^2 - 4x + 4$, set $x^2 - 4x + 4 = 0$.

• Factor: $(x-2)^2 = 0$

• Zero at $x = 2$ (multiplicity 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 $r$ is a zero of even multiplicity, the graph touches the x-axis and turns around at $r$.

• If $r$ is a zero of odd multiplicity, the graph crosses the x-axis at $r$.

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

Example: Multiplicity

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

The Intermediate Value Theorem

The Intermediate Value Theorem states that if $f(a)$ and $f(b)$ have opposite signs for a polynomial function $f$, then there is at least one value $c$ between $a$ and $b$ for which $f(c) = 0$. This guarantees the existence of a real root between $a$ and $b$.

• Useful for locating roots when exact values are difficult to find.

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.

• Turning points help identify the shape and complexity of the graph.

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 noting multiplicities.

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

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

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

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

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

• End behavior: Falls left, rises right.

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

• y-intercept: $f(0) = -24$.

• No symmetry.

• Maximum turning points: 2.

Additional info: Evaluating at $x = 1$ and $x = 2$ gives $f(1) = -36$, $f(2) = -32$ to help scale the graph.