5.3 Higher Order Polynomials
In the previous section we explored the short run behavior of quadratics, a special case of polynomials. In this section we will explore the behavior of polynomials in general. The basic building blocks of polynomials are power functions.
Power Function
A power function is a function that can be represented in the form
Where the base is a variable and the exponent, p, is a number.
Characteristics of Power Functions
Shown to the right are the graphs of , and , all even whole number powers. Notice that all these graphs have a fairly similar shape, very similar to a quadratic, but as the power increases the graphs flatten somewhat near the origin, and become steeper away from the origin.
To describe the behavior as numbers become larger and larger, we use the idea of infinity. The symbol for positive infinity is , and for negative infinity. When we say that “x approaches infinity”, which can be symbolically written as , we are describing a behavior – we are saying that is getting large in the positive direction.
With the even power function, as the input becomes large in either the positive or negative direction, the output values become very large positive numbers. Equivalently, we could describe this by saying that as approaches positive or negative infinity, the values approach positive infinity. In symbolic form, we could write: as , .
Shown here are the graphs of , and , all odd whole number powers. Notice all these graphs look similar, but again as the power increases the graphs flatten near the origin and become steeper away from the origin.
For these odd power functions, as approaches negative infinity, approaches negative infinity. As approaches positive infinity, approaches positive infinity. In symbolic form we write: as , and as , .
Long Run Behavior
The behavior of the graph of a function as the input takes on large negative values () and large positive values () as is referred to as the long run behavior of the function.
Polynomials
Recall our definitions of polynomials from chapter 1.
Terminology of Polynomial Functions
A polynomial is function that can be written as
Each of the constants are called coefficients and can be positive, negative, or zero, and be whole numbers, decimals, or fractions.
A term of the polynomial is any one piece of the sum, that is any . Each individual term is a transformed power function.
The degree of the polynomial is the highest power of the variable that occurs in the polynomial.
The leading term is the term containing the highest power of the variable: the term with the highest degree.
The leading coefficient is the coefficient of the leading term.
Because of the definition of the “leading” term we often rearrange polynomials so that the powers are descending.
For any polynomial the long run behavior of the polynomial will match the long run behavior of the leading term.
Example of Polynomial Graph
What can we determine about the long run behavior and degree of the equation