Polynomial Functions

 

A polynomial function is a finite sum or difference of monomials. (A monomial is a single term with whole numbers as exponents.) This means that the terms of the function are separated by a + or –, and all the powers of the terms are positive integers. The degree of the polynomial is determined by the highest degree term of the polynomial.

 

Example

 

4x2 – 8x +2x5 – 7 is a polynomial of 4 terms of degree 5.

 

 

Polynomial functions have specific names given to them based on the degree. See the table below for a list of these names.

 

Degree

Specific Name

 

1

Linear

2

Quadratic

3

Cubic

4

Quartic

5

Quintic

 

 

         

 

 

Analysis of a Cubic Function Using the Calculator

 

 

Let’s now consider a polynomial function, a cubic, and how the menus on the calculator can help us evaluate the function.

 

F(x) = .33x3 – 1.33x

 

What we are interested in finding are the following: x-intercepts, y-intercept, relative maxima, relative minima, and range of the function. The x-intercepts, y-intercept, relative maxima, and relative minima should be reported as ordered pairs.

 

 

 

The graph of the function is in the above figure. So, after you graph the function by typing it into Y1 and ZOOM 6, your graph should appear like the one in the figure.

 

Finding X-intercepts

 

To find the x-intercepts, ZOOM IN on the curve a little: ZOOM 2 then use the blue arrow key to get the cursor on the zero point where the axes cross. Then press ENTER. The calculator asks for Left Bound? Use the left arrow key and move 2 clicks away from the point; Right Bound? Use the right arrow key and move 4 clicks to get on the right hand side of the point. Guess? Use the left arrow and move until it looks like the cursor is on the point where the curve crosses the x-axis. Then ENTER. Your calculator should tell you the point is (0, 0), the first x-intercept.

 Now follow the same procedure with the point on the left. Left Bound? Move the cursor so it is BELOW the axis. ENTER. Right Bound? Move the cursor so it crosses the x-axis and is above the axis. Guess? Now use the left button to move as close to the axis as you can. ENTER. Your calculator should tell you (– 2, 0) is the left x-intercept.

For the third x-intercept, follow these same steps, and your calculator should report (2, 0) as the third x-intercept. Now you have all three x-intercepts that you see on the graph. I know this is a simple application, but I wanted things easy to follow.

 

Finding Y-intercepts

 

Finding the y-intercept means we want the y-value when x = 0. So, 2ND CALC, VALUE, ENTER. The calculator asks X = ?. Type in 0 because the y-axis means that x = 0.  The point is (0, 0).

 

Finding Relative Maximum

 

The relative maximum is on the left side of the graph in quadrant II. 2ND CALC 4 ENTER,  and the calculator asks Left Bound? Use the left arrow key to move the cursor to the left side of the hump, but as close to the top as you can, ENTER. Right Bound? Use the right arrow key to move to the down side of the hump, but again as close to the top as you can, ENTER. Guess? Arrow to the center of the hump, ENTER. The point should be (– 1.16, 1.03).

 

Finding Relative Minimum

 

The relative minimum is on the right side of the graph in quadrant IV. 2ND CALC 3 ENTER, and the calculator asks Left Bound? Use the left arrow key to move the cursor to the left side of the dip, but as close to the bottom step as you can, ENTER. Right Bound? Use the right arrow key to move to the up side of the dip, but again as close to the bottom step as you can, ENTER. Guess? Arrow to the center of the bottom step, ENTER. The point should be (1.16, –1.03). Notice these points are the same except for the switch in the signs (because of the quadrant change).

 

Finding the Range

 

Look again at the graph in the above figure. Notice that the curve comes in from the bottom left at –∞, and goes out on the right at +∞. This means all the values of y are defined on the function. Therefore, the range is {all reals}.

 

 

 

Finding Solutions for an Equation

 

Suppose we want to solve:    .33x3 – 1.33x = 2.   In Y1 put the left side of the equation, and in Y2 put the right side of the equation, graph using ZOOM 6. Now you may want to ZOOM IN as before. You should see a graph like the graph in the figure below.

 

 

Notice that the line y = 2 intersects our cubic in one point. We are looking for the x-value of that point. 2ND CALC 5 ENTER. Notice that the graph intersection is out of the window. So, change xmin to 0 and xmax to 5 to see the intersection better. 2ND CALC 5 ENTER. Now the calculator asks First Curve? And at the top of your screen you should see the cubic and your cursor should be on the cubic. ENTER. Second Curve? And y = 2 is at the top with the cursor on the line. Use the arrow key to move to as close to the intersection point as possible and press ENTER. The point is (2.53, 2).  OR you can write it in the form X = {2.53}.

 

You may use these instructions on the test.