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Graphing Polynomials

Absolute and local extrema, critical values, end behavior

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Finding and Defining Parts of a Polynomial Function Graph

The prototype for a roller coaster is represented by the equation \begin{align*}y = x^5 - 8x^3 + 10x + 6\end{align*}y=x58x3+10x+6. What is the maximum height the coaster will reach over the domain [-1, 2]?

Parts of a Polynomial Graph

By now, you should be familiar with the general idea of what a polynomial function graph does. It should cross the \begin{align*}x-\end{align*}xaxis as many times as the degree, unless there are imaginary solutions. It will curve up and down and can have a maximum and a minimum. Let’s define the parts of a polynomial function graph here.

Notice that in both the cubic (third degree, on the left) and the quartic (fourth degree, on the right) functions, there is no vertex. We now have minimums and maximums. If there are more than one minimum or maximum, there will be an absolute maximum/minimum, which is the lowest/highest point of the graph. A local maximum/minimum is a maximum/minimum relative to the points around it. The places where the function crosses the \begin{align*}x-\end{align*}xaxis are still the solutions (also called \begin{align*}x-\end{align*}xintercepts, roots or zeros). In the quartic function, there is a repeated root at \begin{align*}x = 4\end{align*}x=4. A repeated root will touch the \begin{align*}x-\end{align*}xaxis without passing through or it can also have a “jump” in the curve at that point (see the first problem below). All of these points together (maximums, minimums, \begin{align*}x-\end{align*}xintercepts, and \begin{align*}y-\end{align*}yintercept) are called critical values.

Another important thing to note is end behavior. It is exactly what it sounds like; how the “ends” of the graph behaves or points. The cubic function above has ends that point in the opposite direction. We say that from left to right, this function is mostly increasing. The quartic function’s ends point in the same direction, both positive, just like a quadratic function. When considering end behavior, look at the leading coefficient and the degree of the polynomial.




Let's use a table to graph \begin{align*}y=x^3\end{align*}y=x3.

Draw a table and pick at least 5 values for \begin{align*}x\end{align*}x.

\begin{align*}x\end{align*}x \begin{align*}x^3\end{align*}x3 \begin{align*}y\end{align*}y
-2 \begin{align*}(-2)^3\end{align*}(2)3 -8
-1 \begin{align*}(-1)^3\end{align*}(1)3 -1
0 \begin{align*}0^3\end{align*}03 0
1 \begin{align*}1^3\end{align*}13 1
2 \begin{align*}2^3\end{align*}23 8

Plot the points and connect. This particular function is the parent graph for cubic functions. Recall from quadratic functions, that the parent graph has a leading coefficient of 1, no other \begin{align*}x-\end{align*}xterms, and no \begin{align*}y-\end{align*}yintercept. \begin{align*}y=x^4\end{align*}y=x4 and \begin{align*}y=x^5\end{align*}y=x5 are also parent graphs.

Now, let's analyze the graph below. Find the critical values, end behavior, and find the domain and range.

First, find the solutions. They appear to be (-2, 0), (1, 0), and (2, 0). Therefore, this function has a minimum degree of 3. However, look at the \begin{align*}y-\end{align*}yintercept. The graph slightly bends between the maximum and minimum. This movement in the graph tells us that there are two imaginary solutions (recall that imaginary solutions always come in pairs). Therefore, the function has a degree of 5. Approximate the other critical values:

maximum: (-1.1, 10)

minimum: (1.5, -1.3)

\begin{align*}y-\end{align*}yintercept: (0, 5)

In general, this function is mostly increasing and the ends go in opposite directions. The domain and range are both all real numbers.

When describing critical values, you may approximate their location. You will later use the graphing calculator to find these values exactly.

Sometime it can be tricky to see if a function has imaginary solutions from the graph. Compare the graph in the previous problem to the cubic function above it. Notice that it is smooth between the maximum and minimum. As was pointed out earlier, the graph from the previous problem bends. Any function with imaginary solutions will have a slightly irregular shape or bend like this one does.

Finally, let's sketch a graph of a function with roots \begin{align*}-4, -3, \frac{1}{2}\end{align*}4,3,12, and 3, has an absolute maximum at (2, 5), and has negative end behavior. This function does not have any imaginary roots.

There are several possible answers for this graph because we are only asking for a sketch. You would need more information to get an exact answer. Because this function has negative end behavior and four roots, we know that it will pass through the \begin{align*}x-\end{align*}xaxis four times and face down. The absolute maximum is located between the roots \begin{align*}\frac{1}{2}\end{align*}12 and 3. Plot these five points and connect to form a graph.


Example 1

Earlier, you were asked to find the maximum height the coaster will reach over the domain [-1, 2]. 

Use a table to graph \begin{align*}x^5 - 8x^3 + 10x + 6\end{align*}x58x3+10x+6.

Draw a table and pick at least 5 values for \begin{align*}x\end{align*}x. Remember that we are dealing only with x values between and including -1 and 2.

\begin{align*}x\end{align*}x \begin{align*}y\end{align*}y
-1 3
0 6
0.5 10.03125
1 9
2 -16

Plot the points and connect.

From your graph you can see that the maximum height the roller coaster reaches is just slightly over 10.

Example 2

Use a table to graph \begin{align*}f(x)=-(x+2)^2(x-3)\end{align*}f(x)=(x+2)2(x3).

This function is in intercept form. Because the factor, \begin{align*}(x + 2)\end{align*}(x+2) is squared, we know it is a repeated root. Therefore, the function should just touch at -2 and not pass through the \begin{align*}x-\end{align*}xaxis. There is also a zero at 3. Because the function is negative, it will be generally decreasing. Think of the slope of the line between the two endpoints. It would be negative. Select several points around the zeros to see the behavior of the graph.

\begin{align*}x\end{align*}x \begin{align*}y\end{align*}y
-4 14
-2 0
0 12
2 16
3 0
4 -36

Example 3

Analyze the graph. Find all the critical values, domain, range and describe the end behavior.

There are three real zeros at approximately -3.5, 1, and 7. Notice the curve between the zeros 1 and 7. This indicated there are two imaginary zeros, making this at least a fifth-degree polynomial. Think about an imaginary horizontal line at \begin{align*}y = 3\end{align*}y=3. This line would touch the graph five times, so there should be five solutions. Next, there is an absolute minimum at (-0.5, -7.5), a local maximum at (2.25, 5), a local minimum at (2.25, 2.25) and an absolute maximum at (5, 6). The \begin{align*}y-\end{align*}yintercept is at (0, -6). The domain and range are both all real numbers and the end behavior is mostly decreasing.

Example 4

Draw a graph of the cubic function with solutions of -6 and a repeated root at 1. This function is generally increasing and has a maximum value of 9.

To say the function is “mostly increasing” means that the slope of the line that connects the two ends (arrows) is positive. Then, the function must pass through (-6, 0) and touch, but not pass through (1, 0). From this information, the maximum must occur between the two zeros and the minimum will be the double root.


Use the given \begin{align*}x-\end{align*}xvalues to make a table and graph the functions below.

  1. \begin{align*}f(x) &= x^3-7x^2+15x-2\\ x &= -2, -1, 0, 1, 2, 3, 4\end{align*}f(x)x=x37x2+15x2=2,1,0,1,2,3,4
  2. \begin{align*}g(x) &= -2x^4 - 11x^3 - 3x^2+37x+35\\ x &= -5, -4, -3, -2, -1, 0, 1, 2\end{align*}g(x)x=2x411x33x2+37x+35=5,4,3,2,1,0,1,2
  3. \begin{align*}y &=2x^3+25x^2+100x+125\\ x &= -7,-6,-5,-4,-3,-2,-1,0\end{align*}yx=2x3+25x2+100x+125=7,6,5,4,3,2,1,0

Make your own table and graph the following functions.

  1. \begin{align*}f(x)=(x+5)(x+2)(x-1)\end{align*}f(x)=(x+5)(x+2)(x1)
  2. \begin{align*}y=x^4\end{align*}y=x4
  3. \begin{align*}y=x^5\end{align*}y=x5
  4. Analyze the graphs of \begin{align*}y=x^2, y=x^3, y=x^4\end{align*}y=x2,y=x3,y=x4, and \begin{align*}y=x^5\end{align*}y=x5. These are all parent functions. What do you think the graph of \begin{align*}y=x^6\end{align*}y=x6 and \begin{align*}y=x^7\end{align*}y=x7 will look like? What can you say about the end behavior of all even functions? Odd functions? What are the solutions to these functions?
  5. Writing How many repeated roots can one function have? Why?

Analyze the graphs of the following functions. Find all critical values, the domain, range, and end behavior.

For questions 13-15, make a sketch of the following real-solution functions.

  1. Draw two different graphs of a cubic function with zeros of -1, 1, and 4.5 and a minimum of -4.
  2. A fourth-degree polynomial with roots of -3.2, -0.9, 1.2, and 8.7, positive end behavior, and a local minimum of -1.7.
  3. A fourth-degree function with solutions of -7, -4, 1, and 2, negative end behavior, and an absolute maximum at \begin{align*}\left(-\frac{11}{2}, \frac{1755}{128}\right)\end{align*}.
  4. Challenge Find the equation of the function from #15.

Answers for Review Problems

To see the Review answers, open this PDF file and look for section 6.13. 

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Absolute Maximum/Minimum

The highest and lowest points of a function are referred to as the absolute maximum and minimum, respectively. When referring to the absolute maximum/minimum value, use the y-value.

Critical Values

The critical values are the x-intercepts of a quadratic function.

End behavior

End behavior is a description of the trend of a function as input values become very large or very small, represented as the 'ends' of a graphed function.

Local Maximum

A local maximum is the highest point relative to the points around it. A function can have more than one local maximum.

Local Minimum

A local minimum is the lowest point relative to the points around it. A function can have more than one local minimum.

Parent Graph

A parent graph is the simplest form of a particular type of graph. All other graphs of this type are usually compared to the parent graph.


A solution to an equation or inequality should result in a true statement when substituted for the variable in the equation or inequality.

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