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.

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.

• There are no gaps or jumps in the graph.

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 values of $|x|$, 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.

• If the degree $n$ is odd:

  • If $a_n > 0$, the graph falls left and rises right.

  • If $a_n < 0$, the graph rises left and falls right.

• If the degree $n$ is even:

  • If $a_n > 0$, the graph rises left and rises right.

  • If $a_n < 0$, the graph falls left and falls 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

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.

Factored Form and Degree

When a polynomial is written in factored form, the degree is the sum of the exponents of each factor.

• Example: $f(x) = -4x^2(x-1)^3(x+5)$ has degree $2+3+1=6$ (even).

• The leading coefficient is the product of the coefficients from each factor.

For $f(x) = -4x^2(x-1)^3(x+5)$, the leading coefficient is -4 (negative), so the graph falls left and falls right.

Application Example: Modeling with Polynomials

Polynomials can be used to model real-world phenomena, such as the number of AIDS cases over time. However, the end behavior may limit the model's applicability.

• If the model predicts negative values for quantities that cannot be negative, it is only valid for a limited time period.

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 x-intercepts.

• Each real root corresponds to an x-intercept on the graph.

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

Example: Finding Zeros

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

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

• Set each factor to zero: $x^2 = 0$, $x-2 = 0$, $x+2 = 0$

• Zeros: $x = 0$ (multiplicity 2), $x = 2$, $x = -2$

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: Multiplicity

For $f(x) = (x+1)^2(x-3)^3$, $x = -1$ is a zero of multiplicity 2 (even), and $x = 3$ is a zero of multiplicity 3 (odd).

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 the number of turning points (maximum is $n-1$).

• Example: $f(x) = 2(x+2)^2(x-3)$

  • 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: These notes expand on the original content by providing definitions, examples, and step-by-step strategies for graphing polynomial functions, ensuring a self-contained and comprehensive study guide for College Algebra students.