Basic Classes of Functions: Learn It 1

  • Identify polynomial degrees and solutions, and graph basic odd and even polynomials
  • Graph a piecewise-defined function
  • Describe how algebraic functions, like polynomials, differ from transcendental functions, like sine and exponential functions
  • Draw the graph of a function after it has been moved up or down, stretched or shrunk, or flipped across an axis

Polynomial Functions

 A polynomial function is any function that can be written in the form

[latex]f(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0[/latex]

 

Polynomials are defined by their degree, which is the highest exponent of the variable x with a non-zero coefficient. The leading coefficient is the coefficient of the term with the highest power.

The simplest polynomial, the zero function [latex]f(x)=0[/latex], has a degree of [latex]0[/latex]. A polynomial of degree [latex]1[/latex] is known as a linear function and can be written as [latex]f(x)=mx+b[/latex], where [latex]m[/latex] is non-zero. If a polynomial’s highest degree term is [latex]2[/latex], it’s called a quadratic function, such as [latex]f(x)=ax^2+bx+c[/latex], with [latex]a[/latex] being non-zero. A polynomial with a degree of 3 is termed cubic, and so forth.

terminology of polynomial functions

Diagram to show what the components of the leading term in a function are. The leading coefficient is a_n and the degree of the variable is the exponent in x^n. Both the leading coefficient and highest degree variable make up the leading term. So the function looks like f(x)=a_nx^n +…+a_2x^2+a_1x+a_0.

 

The degree of the polynomial is the highest power of the variable that occurs in the polynomial; it is the power of the first variable if the function is in general form.

 

The leading term is the term containing the variable with the highest power, also called the term with the highest degree.

 

The leading coefficient is the coefficient of the leading term.

How To: Given a Polynomial Function, Identify the Degree and Leading Coefficient

  1. Find the highest power of [latex]x[/latex] to determine the degree of the function.
  2. Identify the term containing the highest power of [latex]x[/latex] to find the leading term.
  3. The leading coefficient is the coefficient of the leading term.

Identify the degree, leading term, and leading coefficient of the following polynomial functions.

[latex]\begin{array}{l} f\left(x\right)=3+2{x}^{2}-4{x}^{3} \\g\left(t\right)=5{t}^{5}-2{t}^{3}+7t\\h\left(p\right)=6p-{p}^{3}-2\end{array}[/latex]