Polynomial Functions and Their Graphs: Study Notes for College Algebra

Polynomial Functions and Their Graphs

Definition of Polynomial Functions

A polynomial function is an expression of the form $f(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0$, where $a_n \neq 0$ and $n$ is a non-negative integer. The coefficients $a_n, a_{n-1}, \ldots, a_0$ are real numbers.

• Degree: The highest power of $x$ in the polynomial.

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

• Constant Term: The term without $x$ (i.e., $a_0$).

Graphs of Polynomial Functions: Smooth and Continuous

The graph of a polynomial function is always smooth (no sharp corners) and continuous (no breaks or holes). This is due to the nature of polynomials, which are defined for all real numbers.

• Polynomial graphs can have multiple turning points and x-intercepts.

• The number of turning points is at most $n-1$, where $n$ is the degree.

End Behavior of Polynomial Functions

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

• If the degree is even and the leading coefficient is positive, both ends rise.

• If the degree is even and the leading coefficient is negative, both ends fall.

• If the degree is odd and the leading coefficient is positive, left end falls and right end rises.

• If the degree is odd and the leading coefficient is negative, left end rises and right end falls.

The Leading Coefficient Test

The Leading Coefficient Test helps determine the end behavior of a polynomial function:

• Examine the sign of the leading coefficient and the degree of the polynomial.

• Use the rules above to predict how the graph will behave as $x \to \pm \infty$.

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 $x$.

• Each zero may have a multiplicity, which is the number of times it appears as a factor.

Multiplicity and x-Intercepts

The multiplicity of a zero affects how the graph interacts with the x-axis:

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

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

• If the multiplicity is greater than 1, the graph may flatten out near the zero.

The Intermediate Value Theorem

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

• This theorem is useful for proving the existence of real zeros in a given interval.

Turning Points of Polynomial Functions

A turning point is a point where the graph changes direction from increasing to decreasing or vice versa. The maximum number of turning points for a polynomial of degree $n$ is $n-1$.

• Turning points correspond to local maxima and minima.

• They help in sketching the overall shape 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 the graph's end behavior.

2. Find x-intercepts by setting $f(x) = 0$ and solving the resulting polynomial equation. If there is an x-intercept at $r$ as a result of $(x - r)^k$ in the complete factorization of $f(x)$, then:

   • If $k$ is even, the graph touches the x-axis at $r$ and turns around.

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

   • If $k > 1$, the graph flattens out near $(r, 0)$.

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

4. Use symmetry, if applicable, to help draw the graph:

   • y-axis symmetry: $f(-x) = f(x)$

   • Origin symmetry: $f(-x) = -f(x)$

5. Use the fact that the maximum number of turning points of the graph is $n-1$, where $n$ is the degree of the polynomial function, to check whether it is drawn correctly.

Example: Graphing Polynomial Functions

Consider the polynomial $f(x) = 3x^3 - 10x + 9$. To graph this function:

• Apply the Leading Coefficient Test: Degree is 3 (odd), leading coefficient is 3 (positive), so left end falls and right end rises.

• Find zeros by solving $3x^3 - 10x + 9 = 0$.

• Determine multiplicity for each zero.

• Find the y-intercept: $f(0) = 9$.

• Check for symmetry: Since $f(-x) = -3x^3 + 10x + 9 \neq f(x)$ and $\neq -f(x)$, there is no symmetry.

• Maximum number of turning points: $3 - 1 = 2$.

Example: Five-Step Strategy for Graphing

For $f(x) = 2(x + 2)(x - 3)$:

• Degree is 2 (even), leading coefficient is 2 (positive), so both ends rise.

• Zeros at $x = -2$ and $x = 3$, both with multiplicity 1 (graph crosses x-axis at both points).

• y-intercept: $f(0) = 2(0 + 2)(0 - 3) = 2 \times 2 \times (-3) = -12$.

• Maximum number of turning points: $2 - 1 = 1$.

Additional info: The blank graph image is relevant for plotting the polynomial function as part of the five-step strategy example.