3.2 Polynomials and Power Functions: Properties, Graphs, and Applications

Polynomial Functions

Definition and Structure

A polynomial is an algebraic expression that can be written as a sum of terms, each consisting of a variable raised to a non-negative integer power and multiplied by a real coefficient. The general form of a polynomial of degree n is:

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

• Degree: The largest exponent of x that appears in the polynomial.

• Coefficients: The real numbers $a_n, a_{n-1}, \ldots, a_0$.

Power Functions

Power functions are a special type of polynomial defined as $f(x) = x^n$, where n is a positive integer. The behavior of power functions depends on whether n is even or odd.

• Even powers: $f(x) = x^{2k}$ are symmetric about the y-axis and have a parabolic shape.

• Odd powers: $f(x) = x^{2k+1}$ are symmetric about the origin and pass through the origin.

Properties of Power Functions

Property	Even Integer ($n$ even)	Odd Integer ($n$ odd)
Graph Symmetry	Y-axis symmetry	Origin symmetry
End Behavior	Both ends up (if leading coefficient > 0)	Left down, right up (if leading coefficient > 0)
Intercepts	Only at origin	Only at origin

Graphing Polynomial Functions

To graph a polynomial function by hand, follow these steps:

1. Identify the degree and leading coefficient.

2. Find the zeros (solutions) by setting $f(x) = 0$.

3. Determine the multiplicity of each zero.

4. Analyze the end behavior using the leading term.

5. Locate turning points and intercepts.

Example: Graph $f(x) = 1 - x^4$ and $g(x) = \frac{1}{2}(x-1)^4$ by identifying their degree, zeros, and end behavior.

Zeros, Multiplicity, and Turning Points

Zeros (Solutions) of Polynomials

The solutions (or roots) of a polynomial are the x-values that make $f(x) = 0$. These are found by factoring or using the quadratic/cubic formula for lower-degree polynomials.

• If $f(x)$ is a factor of $(x - r)$, then $r$ is a root of $f(x)$.

• Polynomials of degree $n$ have at most $n$ real roots.

Multiplicity of Roots

The multiplicity of a root is the number of times a solution is repeated. For example, in $f(x) = (x-2)^2(x+9)$, the solution $x=2$ has multiplicity 2, and $x=-9$ has multiplicity 1.

• Odd multiplicity: The graph crosses the x-axis at the root.

• Even multiplicity: The graph touches (bounces off) the x-axis at the root.

Turning Points

A turning point is a point where the graph changes direction from increasing to decreasing or vice versa. A polynomial of degree $n$ has at most $n-1$ turning points.

End Behavior of Polynomials

Leading Term and End Behavior

For large values of $|x|$, the behavior of a polynomial $f(x) = a_n x^n + \cdots + a_0$ is dominated by its leading term $a_n x^n$. The end behavior mimics the power function $f(x) = a_n x^n$.

• If $n$ is even and $a_n > 0$, both ends go up.

• If $n$ is even and $a_n < 0$, both ends go down.

• If $n$ is odd and $a_n > 0$, left end down, right end up.

• If $n$ is odd and $a_n < 0$, left end up, right end down.

Example: Describe the end behavior of $f(x) = \frac{1}{2}(x^2 + 5)$ and $g(x) = x^3 + 6(x-5)^4$.

Applications and Data Modeling

Modeling with Polynomials

Polynomials can be used to model real-world data, such as fuel efficiency over time. For example, a cubic function can fit data points relating year to average miles per gallon.

Year	Average Miles per Gallon
1980.1	17.0
1981.2	17.3
1982.3	17.6
1983.4	17.8
1984.5	18.1
1985.6	18.4
1986.7	18.6
1987.8	18.9
1988.9	19.2

By fitting a cubic polynomial to this data, one can predict future values or analyze trends.

Summary Tables

Graphing a Polynomial Function

Step	Description
1	Find the degree and leading coefficient
2	Determine the maximum number of turning points ($n-1$)
3	Find the zeros and their multiplicities
4	Analyze end behavior using the leading term
5	Plot intercepts and turning points
6	Sketch the graph

Practice and Homework

• Practice graphing polynomials by hand and identifying key features.

• Homework: Textbook problems on pages 182 and 183, including exercises #12-54 (odd), #64, #80, and #91-95.

Additional info: These notes cover the essential properties and graphing techniques for polynomial and power functions, including real-world modeling and summary tables for exam review.