To make a fair race between a dragster and a funny car, a scientist devised the following polynomial equation:
In the Quadratic Functions chapter, you used the graphing calculator to graph parabolas. Now, we will expand upon that knowledge and graph higher-degree polynomials. Then, we will use the graphing calculator to find the zeros, maximums and minimums.
Solution: These instructions are for a TI-83 or 84. First, press
To adjust the window, press ZOOM. To get the typical -10 to 10 screen (for both axes), press 6:ZStandard. To zoom out, press ZOOM, 3:ZoomOut, ENTER, ENTER. For this particular function, the window needs to go from -15 to 15 for both
Find the zeros, maximum, and minimum of the function from Example A.
Solution: To find the zeros, press
To find the minimum and maximum, the process is almost identical to finding zeros. Instead of selecting 2:Zero, select 3:min or 4:max. The minimum is (1.33, -14.52) and the maximum is (-2, 4).
Solution: If you decide not to use the calculator, plug in zero for
Using the graphing calculator, press
Intro Problem Revisit If you plug the equation
Graph and find the critical values of the following functions.
3. Find the domain and range of the previous two functions.
4. Describe the types of solutions, as specifically as possible, for question 2.
Use the steps given in Examples
1. zeros: -5.874, -2.56, 0.151, 5.283
minimum: (-1.15, -18.59)
local maximum: (-4.62, 40.69)
absolute maximum: (3.52, 113.12)
2. zeros: -1.413, -0.682, 0.672
minimum: (-1.11, 4.41)
maximum: (0.08, -8.12)
4. There are three irrational solutions and two imaginary solutions.
Graph questions 1-6 on your graphing calculator. Sketch the graph in an appropriate window. Then, find all the critical values, domain, range, and describe the end behavior.
- What are the types of solutions in #2?
- Find the two imaginary solutions in #3.
- Find the exact values of the irrational roots in #5.
Determine if the following statements are SOMETIMES, ALWAYS, or NEVER true. Explain your reasoning.
- The range of an even function is
(−∞,max], where max is the maximum of the function.
- The domain and range of all odd functions are all real numbers.
- A function can have exactly three imaginary solutions.
nthdegree polynomial has nreal solutions.
- The parent graph of any polynomial function has one zero.
Challenge The exact value for one of the zeros in #2 is
−4+7√. What is the exact value of the other root? Then, use this information to find the imaginary roots.