The graphing calculator program that comes with every Power Macintosh computer is a very powerful tool for doing algebra. This activity will help use this program to explore the graphs of polynomials.

When you start the graphing calculator, you should see a window similar to this:

You type expressions and equations in the top part of the window. Graphs appear in the bottom part of the window. The buttons on the lower left allow you to zoom in and out. You can see different parts of the graph by dragging the axes with your mouse.

In this exploration, you will explore the graphs of 3rd and 4th degree polynomials. You will use the graphing calculator's animation capability to investigate how each term affects the graph. As you answer the questions that follow, describe what is happening with the graph as you vary the coefficient, but also try to describe why this is happening.

These are very open ended questions that will require you to experiment and try different things. Feel free to zoom in or zoom out, or try values beyond the suggested range, or anything else that you think might help you to clarify your understanding of why the graphs are behaving the way they are.

Type the equation y = x^{3} + x^{2} + x into the
graphing calculator. You should see a graph like this:

You can use the zoom buttons in the lower left to zoom in or out. You can click and drag the axes to see other parts of the graph. Click on the graph itself to turn on tracing:

You can click and drag the box to get the coordinates of any point along the graph. The box will "stick" for a moment and make a clicking noise at important points such as the x and y intercepts.

Type the equation y = nx^{3} + x^{2} + x into the
graphing calculator. The animation controller will appear at the
bottom of your window. Click the n and set the lowest value to -2,
the highest value to 2, and set the number of steps to 20. Click OK.
Use your mouse to click and drag the slider back and forth. How does
moving the slider affect your graph? What role does the n play in
this equation?

Type the equation y = x^{3} + nx^{2} + x into the
graphing calculator. Click the n on the animation controller and set
the lowest value to -3, the highest value to 3, and the number of
steps to 20. Click OK. Use your mouse to click and drag the slider
back and forth. How does this affect your graph? What role does the n
play in this equation?

Type the equation y = x^{3} + x^{2} + nx into the
graphing calculator. Click the n on the animation controller and set
the lowest value to -2, the highest value to 2, and the number of
steps to 20. Click OK. Use your mouse to click and drag the slider
back and forth. How does this affect your graph? What role does the n
play in this equation?

Type the equation y = x^{4} + x^{3} +
x^{2} + x into the graphing calculator. You should see a
graph like this:

Type the equation y = nx^{4} + x^{3} +
x^{2} + x into the graphing calculator. Click the n on the
animation controller and set the lowest value to -2, the highest
value to 2, and the number of steps to 20. Click OK. Use your mouse
to click and drag the slider back and forth. How does this affect
your graph? What role does the n play in this equation?

Type the equation y = x^{4} + nx^{3} +
x^{2} + x into the graphing calculator. Click the n on the
animation controller and set the lowest value to -3, the highest
value to 3, and the number of steps to 20. Click OK. Use your mouse
to click and drag the slider back and forth. How does this affect
your graph? What role does the n play in this equation?

Type the equation y = x^{4} + x^{3} +
nx^{2} + x into the graphing calculator. Click the n on the
animation controller and set the lowest value to -3, the highest
value to 3, and the number of steps to 20. Click OK. Use your mouse
to click and drag the slider back and forth. How does this affect
your graph? What role does the n play in this equation?

Type the equation y = x^{4} + x^{3} +
x^{2} + nx into the graphing calculator. Click the n on the
animation controller and set the lowest value to -3, the highest
value to 3, and the number of steps to 20. Click OK. Use your mouse
to click and drag the slider back and forth. How does this affect
your graph? What role does the n play in this equation?