Polynomial Functions and Their Graphs: Study Notes for Precalculus

Polynomial Functions and Models

Definition and Degree of Polynomial Functions

A polynomial function 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 real numbers.

• Degree $n$ is a nonnegative integer and is the highest power of $x$ with a nonzero coefficient.

• Domain: All real numbers ($\mathbb{R}$).

• The graph of a polynomial function is always smooth and continuous (no breaks, holes, or sharp corners).

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

Identifying Polynomial Graphs

• Polynomial graphs are smooth and continuous.

• They do not have jumps, holes, or sharp corners.

Note: Only graphs that are smooth and continuous without holes or sharp turns can represent polynomial functions.

Zeros of Polynomial Functions and Their Multiplicity

Definition of a Zero

• If $a$ is a zero of $f(x)$, then:

  • $x = a$ is a solution to $f(x) = 0$

  • $(a, 0)$ is an x-intercept of the graph

  • $(x - a)$ is a factor of $f(x)$

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

Multiplicity of Zeros

• If $(x-a)^n$ is a factor and $(x-a)^{n+1}$ is not, then $a$ is a zero of multiplicity $n$.

Multiplicity $n$	Sign Change	Graph Behavior at $x=a$
1	Changes sign	Crosses the x-axis
Even	Same sign	Touches the x-axis
Odd ($\geq 3$)	Changes sign	Touches & crosses the x-axis (flattens)

Turning Points and End Behavior

Turning Points

• A turning point is where the graph changes direction (from increasing to decreasing or vice versa).

• If $f$ is a polynomial of degree $n$, it has at most $n-1$ turning points.

• If a graph has $n-1$ turning points, the degree is at least $n$.

End Behavior of Polynomial Functions

The end behavior of $f(x) = a_n x^n + \cdots + a_0$ is determined by the leading term $a_n x^n$.

Degree $n$	Leading Coefficient $a_n$	End Behavior as $x \to \infty$	End Behavior as $x \to -\infty$
Even	Positive	$f(x) \to \infty$	$f(x) \to \infty$
Even	Negative	$f(x) \to -\infty$	$f(x) \to -\infty$
Odd	Positive	$f(x) \to \infty$	$f(x) \to -\infty$
Odd	Negative	$f(x) \to -\infty$	$f(x) \to \infty$

Analyzing the Graph of a Polynomial Function

Step-by-Step Analysis

1. Find the degree of the polynomial.

2. Find the x- and y-intercepts by solving $f(x) = 0$ and evaluating $f(0)$.

3. Find zeros and their multiplicities to determine if the graph crosses or touches the x-axis at each zero.

4. Determine the number of turning points (at most $n-1$ for degree $n$).

5. Analyze end behavior using the leading term.

6. Sketch the graph using all the above information.

Example: $f(x) = (x-1)(x+1)^2(x-3)$

• Degree: $1 + 2 + 1 = 4$ (quartic)

• Zeros: $x=1$ (mult. 1, crosses), $x=-1$ (mult. 2, touches), $x=3$ (mult. 1, crosses)

• Turning points: At most $4-1=3$

• End behavior: Leading coefficient positive, even degree: both ends up

Extra Exercises and Applications

• Identify which graphs could be polynomial functions and state the least possible degree.

• Construct polynomial functions from given graphs and intercepts.

• Analyze polynomial functions using the six-step process above.

Summary Table: Key Properties of Polynomial Functions

Property	Description
Degree	Highest power of $x$ with a nonzero coefficient
Zeros	Values of $x$ where $f(x) = 0$
Multiplicity	Number of times a zero is repeated
Turning Points	At most $n-1$ for degree $n$
End Behavior	Determined by leading term $a_n x^n$
Graph Shape	Smooth and continuous

Additional info: These notes are based on standard precalculus curriculum and include expanded explanations, examples, and tables for clarity.