Parts of a Polynomial Graph
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.
Now, let's analyze the graph below. Find the critical values, end behavior, and find the domain and range.
maximum: (-1.1, 10)
minimum: (1.5, -1.3)
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.
Earlier, you were asked to find the maximum height the coaster will reach over the domain [-1, 2].
Plot the points and connect.
From your graph you can see that the maximum height the roller coaster reaches is just slightly over 10.
Analyze the graph. Find all the critical values, domain, range and describe the end behavior.
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.
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
y=x2,y=x3,y=x4, and y=x5. These are all parent functions. What do you think the graph of y=x6and y=x7will 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 13-15, make a sketch of the following real-solution functions.
- Draw two different graphs of a cubic function with zeros of -1, 1, and 4.5 and a minimum of -4.
- 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.
- A fourth-degree function with solutions of -7, -4, 1, and 2, negative end behavior, and an absolute maximum at
- 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.