To make a fair race between a dragster and a funny car, a scientist devised the following polynomial equation:
. What is the maximum point of this function's graph?
James Sousa: Ex: Solve a Polynomial Equation Using a Graphing Calculator (Approximate Solutions)
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.
using a graphing calculator.
These instructions are for a TI-83 or 84
. First, press
. If there are any functions in this window, clear them out by highlighting the = sign and pressing
, enter in the polynomial. It should look like:
To adjust the window, press
. To get the typical -10 to 10 screen (for both axes), press
To zoom out, press
ZOOM, 3:ZoomOut, ENTER, ENTER.
For this particular function, the window needs to go from -15 to 15 for both
. To manually input the window, press WINDOW and change the
so that you can see the zeros, minimum and maximum. Your graph should look like the one to the right.
Find the zeros, maximum, and minimum of the function from Example A.
To find the zeros, press
to get the
and you will be asked “Left Bound?” by the calculator. Move the cursor (by pressing the
) so that it is just to the left of one zero. Press
Then, it will ask “Right Bound?” Move the cursor just to the right of that zero. Press
The calculator will then ask “Guess?” At this point, you can enter in what you think the zero is and press
again. Then the calculator will give you the exact zero. For the graph from Example A, you will need to repeat this three times. The zeros are -2.83, -1, and 2.83.
To find the minimum and maximum, the process is almost identical to finding zeros. Instead of selecting
. The minimum is (1.33, -14.52) and the maximum is (-2, 4).
intercept of the graph from Example A.
If you decide not to use the calculator, plug in zero for
and solve for
Using the graphing calculator, press
to get the
shows up at the bottom of the screen. If there is a value there, press
to remove it. Then press
The calculator should then say “
Intro Problem Revisit
If you plug the equation
into your calculator, you find that the maximum occurs when
. At that value of
equals 1754.43. Therefore the maximum point of the function's graph is (6.15105, 1754.43).
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
intercept: (0, -4)
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
intercept: (0, -8)
minimum: (-1.11, 4.41)
maximum: (0.08, -8.12)
3. The domain of #1 is all real numbers and the range is all real numbers less than the maximum;
. The domain and range of #2 are all real numbers.
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.
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
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.
degree polynomial has
The parent graph of any polynomial function has one zero.
The exact value for one of the zeros in #2 is
. What is the exact value of the other root? Then, use this information to find the imaginary roots.