6.13: Finding and Defining Parts of a Polynomial Function Graph
The prototype for a roller coaster is represented by the equation
Watch This
First watch this video.
James Sousa: Ex: Increasing/ Decreasing/ Relative Extrema from Analyzing a Graph
Then watch this video.
James Sousa: Summary of End Behavior or Long Run Behavior of Polynomial Functions
Guidance
By now, you should be familiar with the general idea of what a polynomial function graph does. It should cross the
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
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.
Example A
Use a table to graph
Solution: Draw a table and pick at least 5 values for




2 

8 
1 

1 
0 

0 
1 

1 
2 

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
Example B
Analyze the graph below. Find the critical values, end behavior, and find the domain and range.
Solution: 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
maximum: (1.1, 10)
minimum: (1.5, 1.3)
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. In the next concept, we will 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 Example B to the cubic function above. Notice that it is smooth between the maximum and minimum. As was pointed out earlier, the graph from Example B bends. Any function with imaginary solutions will have a slightly irregular shape or bend like this one does.
Example C
Sketch a graph of a function with roots
Solution: 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
Intro Problem Revisit
Use a table to graph
Draw a table and pick at least 5 values for



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.
Guided Practice
1. Use a table to graph
2. Analyze the graph. Find all the critical values, domain, range and describe the end behavior.
3. 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.
Answers
1. This function is in intercept form. Because the factor,



4  14 
2  0 
0  12 
2  16 
3  0 
4  36 
2. 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 fifthdegree polynomial. Think about an imaginary horizontal line at
3. 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.
Vocabulary
 Absolute Maximum/Minimum

The highest/lowest point of a function. When referring to the absolute maximum/minimum value, use the
y− value.
 Local Maximum/Minimum
 The highest/lowest point relative to the points around it. A function can have multiple local maximums or minimums.
 Solutions

The
x− intercepts. Also called roots or zeros.
 Critical Values

The
x− intercepts, maximums, minimums, andy− intercept.
 End Behavior
 How the ends of a graph look. End behavior depends on the degree of the function and the leading coefficient.
 Parent Graph

The most basic function of a particular type. It has a leading coefficient of 1, no additional
x− terms, and no constant.
Practice
Use the given

f(x)x=x3−7x2+15x−2=−2,−1,0,1,2,3,4 
g(x)x=−2x4−11x3−3x2+37x+35=−5,−4,−3,−2,−1,0,1,2 
yx=2x3+25x2+100x+125=−7,−6,−5,−4,−3,−2,−1,0
Make your own table and graph the following functions.

f(x)=(x+5)(x+2)(x−1) 
y=x4 
y=x5  Analyze the graphs of \begin{align*}y=x^2, y=x^3, y=x^4\end{align*}, and \begin{align*}y=x^5\end{align*}. These are all parent functions. What do you think the graph of \begin{align*}y=x^6\end{align*} and \begin{align*}y=x^7\end{align*} will look like? What can you say about the end behavior of all even functions? Odd functions? What are the solutions to these functions?
 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 1315, make a sketch of the following realsolution functions.
 Draw two different graphs of a cubic function with zeros of 1, 1, and 4.5 and a minimum of 4.
 A fourthdegree polynomial with roots of 3.2, 0.9, 1.2, and 8.7, positive end behavior, and a local minimum of 1.7.
 A fourthdegree 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*}.
 Challenge Find the equation of the function from #15.
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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 value.Critical Values
The critical values are the 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.solution
A solution to an equation or inequality should result in a true statement when substituted for the variable in the equation or inequality.Image Attributions
Here you'll learn about the different parts of graphs for higherdegree polynomials.