Identify the x-Intercepts of Polynomial Functions whose Equations are Factorable
This section will dig deeper into the relationship between the graph of a polynomial function and its equation. You will see how to use the factors of a polynomial function to determine where the x-intercepts are, and you will also learn about the multiplicity of a zero (x-intercept) and how to find it.
Recognizing Characteristics of Graphs of Polynomial Functions
Polynomial functions of degree \(2\) or more have graphs that do not have sharp corners; recall that these types of graphs are called smooth curves. Polynomial functions also display graphs that have no breaks. Curves with no breaks are called continuous. Figure 1 shows a graph that represents a polynomial function and a graph that represents a function that is not a polynomial.
Figure 1
EXAMPLE 1
Recognizing Polynomial Functions
Which of the graphs in Figure 2 represents a polynomial function?
Figure 2
Solution
The graphs of \(f\) and \(h\) are graphs of polynomial functions. They are smooth and continuous.
The graphs of \(g\) and \(k\) are graphs of functions that are not polynomials. The graph of function \(g\) has a sharp corner. The graph of function \(k\) is not continuous.
Q&A
Do all polynomial functions have as their domain all real numbers?
Yes. Any real number is a valid input for a polynomial function.