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Graphs of Polynomials Using Zeros

x- and y-intercepts, signs, and leading term tests used to sketch a graph.

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Graphs of Polynomials Using Zeros

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?

Watch This

This video is a good introduction to graphing cubic and higher degree polynomials. Note that a graphing calculator is used in the video.

Embedded Video:

- James Sousa: Graphing Cubic Functions

Guidance

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:

Example A

Find the roots (zeroes) of the polynomial:

Solution:

Start by factoring:

To find the zeros, set h(x)=0 and solve for x.

This gives

or

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.

Example B

Find the zeros of .

Solution

The polynomial can be written as

To solve the equation, we simply set it equal to zero

this gives

or

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 .

Example C

Graph the polynomial function .

Solution

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 .

This gives

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.

In the introduction to the lesson, it was noted that there are similarities in graphing using zeroes between quadratic functions and higher-degree polynomials. Were you able to identify some of those similarities?

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.

Vocabulary

Cubic Function: A function containing an term as the highest power of x.

Quartic Function: A function containing an term as the highest power of x.

Zeroes of a Polynomial: The values output by the function (the values of or ), when the input (the value) is zero, or vice-versa.

Interval: A portion of a function, generally defined by a starting and ending value of .

Guided Practice

Questions

Sketch a graph of each power function using the properties of the power functions.

1)

2)

3)

4) Find the zeros and sketch a graph of the polynomial

5) Graph

Solutions

1) Solution:
Step 1: By applying the leading term test, we can say that since the co-efficient is , and since the power is even, the end behavior of the graph resembles:
Step 2: By solving the equation for and , we get the points: and .
Step 3: This suggests
2) Solution:
Step 1: By applying the leading term test, we can say that since the co-efficient is , and since the power is odd, the end behavior of the graph resembles:
Step 2: By solving the equation for and , we get the points: and .
Step 3: This suggests
3) Solution:
Step 1: By applying the leading term test, we can say that since the co-efficient is , and since the power is even, the end behavior of the graph resembles:
Step 2: By solving the equation for and , we get the points: and .
Step 3: This suggests

4) This is a factorable equation,

Setting ,

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.

5) 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
-6 320 + Above
-5 0 NA
(-5, -1) -2 144 + Above
-1 0 NA
(-1, 2) 0 -100 - Below
2 0 NA
3 -256 - Below

Finally, use this information and the test points to sketch a graph of .

Practice

  1. If c is a zero of f, then c is a/an _________________________ of the graph of f
  2. If c is a zero of f, then (x - c) is a factor of ___________________?
  3. Find the zeros of the polynomial:

Consider the function: .

  1. How many zeros (x-intercepts) are there?
  2. 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 #s 9-12

  1. Given: State: a) The leading term: b) The degree of the polynomial: c) The leading coefficient:

Determine the equation of the polynomial based on the graph:

Vocabulary

Cubic Function

Cubic Function

A cubic function is a function containing an x^{3} term as the highest power of x.
Intercept

Intercept

The intercepts of a curve are the locations where the curve intersects the x and y axes. An x intercept is a point at which the curve intersects the x-axis. A y intercept is a point at which the curve intersects the y-axis.
interval

interval

An interval is a specific and limited part of a function.
Leading-Term Test

Leading-Term Test

The leading-term test is a test to determine the end behavior of a polynomial function.
Polynomial

Polynomial

A polynomial is an expression with at least one algebraic term, but which does not indicate division by a variable or contain variables with fractional exponents.
Polynomial Graph

Polynomial Graph

A polynomial graph is the graph of a polynomial function. The term is most commonly used for polynomial functions with a degree of at least three.
Quartic Function

Quartic Function

A quartic function is a function f(x) containing an x^{4} term as the highest power of ''x''.
Roots

Roots

The roots of a function are the values of x that make y equal to zero.
Zeroes

Zeroes

The zeroes of a function f(x) are the values of x that cause f(x) to be equal to zero.

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