Polynomial Functions and Their Graphs: Study Guide

Polynomial Functions and Their Graphs

Definition of a Polynomial Function

A polynomial function is an algebraic expression consisting of terms in the form $a_n x^n$, where the exponents are non-negative integers and the coefficients are real numbers. The general form is:

• $f(x) = a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0$

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

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

• All coefficients $a_n, a_{n-1}, \ldots, a_0$ are real numbers.

• If $f(x) = c$, the degree is 0 (constant function).

• If $f(x) = mx + b$, the degree is 1 (linear function).

• Focus is on polynomials of degree 3 or higher.

Smooth and Continuous Graphs

Polynomial functions of degree 2 or higher have graphs that are smooth (no sharp corners) and continuous (no breaks or holes).

• Smooth: Only rounded curves, no abrupt changes in direction.

• Continuous: Can be drawn without lifting the pencil.

• Graphs with discontinuities or sharp points are not polynomial graphs.

End Behavior of Polynomial Functions

The end behavior describes how the graph behaves as $x$ approaches positive or negative infinity. This is determined by the degree and leading coefficient.

• As $x \to \infty$ or $x \to -\infty$, the graph will rise or fall without bound.

• End behavior is summarized by the Leading Coefficient Test.

Leading Coefficient Test

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

• If $n$ is odd:

  • If $a_n > 0$: Graph falls left, rises 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 left, falls 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 both 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 both left and right. $\lim_{x \to -\infty} f(x) = -\infty$ $\lim_{x \to \infty} f(x) = -\infty$ Example: $f(x) = -x^2$

Examples of End Behavior

• $f(x) = 2x^3 - 11x^2 + 6x + 3$ (degree 3, leading coefficient 2): Odd degree, positive leading coefficient. Graph falls left, rises right.

• $f(x) = -x^4 - 2x^2 + 11x + 6$ (degree 4, leading coefficient -1): Even degree, negative leading coefficient. Graph falls 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 $x$.

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

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

Multiplicities of Zeros

When factoring a polynomial, each factor produces a zero. The multiplicity of a zero is the number of times its corresponding factor appears.

• If a root $r$ has odd multiplicity, the graph crosses the x-axis at $r$.

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

• Example: $f(x) = x^3 + 2x^2 - 4x - 8$ factors to $(x + 2)^2(x - 2)$.

  • Zero at $x = -2$ (multiplicity 2, graph bounces).

  • Zero at $x = 2$ (multiplicity 1, graph crosses).

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 value $c$ between $a$ and $b$ where $f(c) = 0$.

• This guarantees at least one real root between $a$ and $b$.

• 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. 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.

Five-Step Strategy for Graphing Polynomials

To graph a polynomial function, follow these steps:

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

2. Find x-intercepts and determine where the graph crosses or bounces on the x-axis.

   • If root $r$ has even multiplicity: graph touches and turns at $r$.

   • If root $r$ has odd multiplicity: graph crosses at $r$.

   • If multiplicity $> 1$: graph flattens at $(r, 0)$.

3. Find the y-intercept (set $x = 0$).

4. Use symmetry if applicable:

   • If $f(-x) = f(x)$: y-axis symmetry (even function).

   • If $f(-x) = -f(x)$: origin symmetry (odd function).

5. Check turning points: Ensure the graph has at most $n - 1$ turning points.

Examples: Graphing Polynomial Functions

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

• Graph $f(x) = x^4 - 6x^3 + 9x^2$ using the five-step strategy.

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

Summary Table: End Behavior by Degree and Leading Coefficient

Degree ($n$)	Leading Coefficient ($a_n$)	End Behavior (Left)	End Behavior (Right)
Odd	Positive	Falls ($-\infty$)	Rises ($\infty$)
Odd	Negative	Rises ($\infty$)	Falls ($-\infty$)
Even	Positive	Rises ($\infty$)	Rises ($\infty$)
Even	Negative	Falls ($-\infty$)	Falls ($-\infty$)

Additional info: Examples and applications were expanded for clarity. The summary table was inferred for completeness and exam preparation.