How is finding and using the zeroes of a higher-degree polynomial related to the same process you have used in the past on quadratic functions?
Graphing Polynomials Using Zeros
The following procedure can be followed when graphing a polynomial function.
- Use the leading-term test to determine the end behavior of the graph.
- Find the intercept(s) of by setting and then solving for .
- Find the intercept of by setting and finding .
- Use the intercept(s) to divide the axis into intervals and then choose test points to determine the sign of on each interval.
- Plot the test points.
- If necessary, find additional points to determine the general shape of the graph.
The Leading-Term Test
If is the leading term of a polynomial. Then the behavior of the graph as or can be known by one the four following behaviors:
1. If and even:
2. If and even:
3. If and odd:
4. If and odd:
Earlier, you were asked to identify some similarities in graphing using zeroes between quadratic functions and higher-degree polynomials.
Despite the more complex nature of the graphs of higher-degree polynomials, the general process of graphing using zeroes is actually very similar. In both cases, your goal is to locate the points where the graph crosses the x or y axis. In both cases, this is done by setting the y value equal to zero and solving for x to find the x axis intercepts, and setting the x value equal to zero and solving for y to find the y axis intercepts.
Find the roots (zeroes) of the polynomial:
Start by factoring:
To find the zeros, set h(x)=0 and solve for x.
So we say that the solution set is . They are the zeros of the function . The zeros of are the intercepts of the graph below.
Find the zeros of .
The polynomial can be written as
To solve the equation, we simply set it equal to zero
Notice the occurrence of the zeros in the function. The factor occurred twice (because it was squared), the factor occurred once and the factor occurred three times. We say that the zero we obtain from the factor has a multiplicity and the factor has a multiplicity .
Graph the polynomial function .
Since the leading term here is then , and even. Thus the end behavior of the graph as and is that of Box #2, item 2.
We can find the zeros of the function by simply setting and then solving for .
So we have two intercepts, at and at , with multiplicity for and multiplicity for .
To find the intercept, we find , which gives
So the graph passes the axis at .
Since the intercepts are 0 and , they divide the axis into three intervals: and . Now we are interested in determining at which intervals the function is negative and at which intervals it is positive. To do so, we construct a table and choose a test value for from each interval and find the corresponding at that value.
|Interval||Test Value||Sign of||Location of points on the graph|
|-1||-5||-||below the axis|
|+||above the axis|
|1||-1||-||below the axis|
Those test points give us three additional points to plot: , and (1, -1). Now we are ready to plot our graph. We have a total of three intercept points, in addition to the three test points. We also know how the graph is behaving as and . This information is usually enough to make a rough sketch of the graph. If we need additional points, we can simply select more points to complete the graph.
Find the zeros and sketch a graph of the polynomial
This is a factorable equation,
the first term gives
and the second term gives
So the solutions are and , a total of four zeros of . Keep in mind that only the real zeros of a function correspond to the intercept of its graph.
Use the zeros to create a table of intervals and see whether the function is above or below the axis in each interval:
|Interval||Test value||Sign of||Location of graph relative to axis|
Finally, use this information and the test points to sketch a graph of .
- If c is a zero of f, then c is a/an _________________________ of the graph of f.
- If c is a zero of f, then (x - c) is a factor of ___________________?
- Find the zeros of the polynomial:
Consider the function: .
- How many zeros (x-intercepts) are there?
- What is the leading term?
Find the zeros and graph the polynomial. Be sure to label the x-intercepts, y-intercept (if possible) and have correct end behavior. You may use technology for questions 9-12.
- The leading term:
- The degree of the polynomial:
- The leading coefficient:
Determine the equation of the polynomial based on the graph:
To see the Review answers, open this PDF file and look for section 2.4.