A large screen applet helps you explore graphical properties of third order polynomials of the form
f(x) = ax3 + bx + c
where a, b and c are parameters that may be changed in order to investigate their effects on the graph of the polynomial.
1 - Click on the button "click here to start", above, to start the applet and maximize the window obtained.
2 - Change parameters a, b and c. Set a to 0 and b to non zero values; what is the graph of the polynomial obtained? Explain.
3 - Set c to 0 and change parameters a and b to non zero values. Is the polynomial obtained odd? For what values of a and b is the polynomial odd when c is equal to 0?
4 - Set c to a non zero value. Set a and b to non zero values is the polynomial obtained odd?
5 - Change parameters a, b and c. What is the minimum number of x intercepts of the graph of the polynomial? What is the maximum number of x intercepts? Can the polynomial have only two x intercepts?
6 - Set a to a positive value. What happens to f(x) as x increases? What happens to f(x) as x decreases?
7 - Set a to a negative value. What happens to f(x) as x increases? What happens to f(x) as x decreases?
More references and links to polynomial functions.
Derivatives of Polynomial Functions.
Polynomial Functions, Zeros, Factors and Intercepts
Find Zeros of Polynomial Functions - ProblemsGraphs of Polynomial Functions - Questions.