Polynomial Functions and Their Graphs: Definitions, Properties, and Graphing Strategies

Polynomial Functions and 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 all coefficients $a_n, a_{n-1}, \ldots, a_0$ are real numbers.

• Degree: The highest power of $x$ (i.e., $n$) is called the degree of the polynomial.

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

• Examples:

  • Constant function: $f(x) = c$ (degree 0)

  • Linear function: $f(x) = mx + b$ (degree 1)

  • We focus on polynomials of degree 3 or higher in this section.

Smooth and Continuous Graphs

The graph of a polynomial function of degree 2 or higher is always smooth (no sharp corners) and continuous (no breaks or holes).

• Smooth: The graph consists of rounded curves only.

• Continuous: The graph can be drawn without lifting your pencil.

• Non-polynomial graphs may have discontinuities (holes, jumps, or asymptotes) or sharp corners.

End Behavior of Polynomial Functions

End behavior describes how the values of $f(x)$ behave as $x$ approaches positive or negative infinity. The end behavior is determined by the degree and leading coefficient of the polynomial.

The Leading Coefficient Test

• Given $f(x) = a_n x^n + \ldots + a_0$:

• If $n$ is odd:

  • If $a_n > 0$: Graph falls to the left and rises to the right.

    • $\lim_{x \to -\infty} f(x) = -\infty$

    • $\lim_{x \to \infty} f(x) = \infty$

    • Example: $f(x) = x^3$

  • If $a_n < 0$: Graph rises to the left and falls to the right.

    • $\lim_{x \to -\infty} f(x) = \infty$

    • $\lim_{x \to \infty} f(x) = -\infty$

    • Example: $f(x) = -x^3$

• If $n$ is even:

  • If $a_n > 0$: Graph rises to both the left and right.

    • $\lim_{x \to -\infty} f(x) = \infty$

    • $\lim_{x \to \infty} f(x) = \infty$

    • Example: $f(x) = x^2$

  • If $a_n < 0$: Graph falls to both the left and right.

    • $\lim_{x \to -\infty} f(x) = -\infty$

    • $\lim_{x \to \infty} f(x) = -\infty$

    • Example: $f(x) = -x^2$

Examples:

• For $f(x) = 2x^3 + 11x^2 + 6x + 3$ (degree 3, leading coefficient 2):

  • Odd degree, positive leading coefficient: falls left, rises right.

• For $f(x) = -x^4 - 2x^2 + 11x + 6$ (degree 4, leading coefficient -1):

  • Even degree, negative leading coefficient: falls both left and right.

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 $x$ (often by factoring).

• Example: Find all zeros of $f(x) = x^3 + 2x^2 - 4x - 8$.

Multiplicities of Zeros

When a polynomial is factored, the multiplicity of a zero is the number of times its corresponding factor appears.

• If $(x - r)^k$ is a factor, then $r$ is a zero of multiplicity $k$.

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

• If the multiplicity is even, the graph touches the x-axis and turns around at $r$ (the graph "bounces").

• Example: $f(x) = (x + 2)^2(x - 2)$

  • Zero at $x = -2$ (multiplicity 2, touches x-axis)

  • Zero at $x = 2$ (multiplicity 1, crosses x-axis)

The Intermediate Value Theorem

The Intermediate Value Theorem states that if $f$ is a polynomial function with real coefficients, and $f(a)$ and $f(b)$ have opposite signs, then there is at least one real zero between $a$ and $b$.

• This theorem guarantees the existence of a real root in an interval where the function changes sign.

• Example: Show that $f(x) = x^3 - 3x + 10$ has a real zero between $x = -3$ and $x = -2$.

Turning Points of Polynomial Functions

Turning points are points where the graph changes from increasing to decreasing or vice versa.

• A polynomial of degree $n$ has at most $n - 1$ turning points.

• Turning points help determine the overall shape of the graph.

Five-Step Strategy for Graphing Polynomial Functions

To sketch the graph of a polynomial function, follow these steps:

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

2. Find x-intercepts (zeros) and determine whether the graph crosses or bounces at each zero:

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

   • If the zero has odd multiplicity, the graph crosses the x-axis.

   • If multiplicity is greater than 1, the graph flattens out at the intercept.

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

4. Use symmetry if applicable:

   • If $f(-x) = f(x)$, the graph is symmetric about the y-axis (even function).

   • If $f(-x) = -f(x)$, the graph is symmetric about the origin (odd function).

5. Check the number of turning points (should not exceed $n-1$ for degree $n$).

Example Applications

• Graph $f(x) = x^3 - x^2 - 3x$ using the five-step strategy.

• Graph $f(x) = x^4 - 6x^2 + 9$ using the five-step strategy.

• Graph $f(x) = (x - 3)^2(x + 1)^3$ using the five-step strategy.

• For each, identify end behavior, zeros and their multiplicities, y-intercept, symmetry, and turning points.

Additional info: For more complex polynomials, graphing calculators or software can be used to approximate zeros and turning points.