Education Technology

Interesting Properties of Cubic Functions

Published on 07/30/2005

Activity Overview

This Computer Algebra System (CAS) activity encourages students to investigate numerical and graphical properties of cubic functions, and to verify the results using CAS.

Before the Activity

  • Install TI Connect™ using the TI connectivity cable
  • See the attached PDF file for detailed instructions for this activity
  • Print the attached PDF file for the class
  • During the Activity

    Distribute the page to the class.

    Follow the Activity procedures:

  • Enter a cubic function and graph it
  • Draw a tangent at a point on the curve, and find the point at which the tangent line intersects the curve
  • Explore the relationship between the x-coordinates of the two points, between the slopes of the curve at those points, and between the y-intercepts of the tangent lines at those points

  • Graph a cubic function and understand that it is symmetric to its point of inflection
  • Find the coordinates of the inflection point
  • Translate the function so that the inflection point is at the origin
  • Establish that symmetry exists by proving the translated function is odd

  • Consider a cubic polynomial function with three distinct real zeros
  • Find the point where the tangent line drawn to the curve at the average of two of the three zeros intersects the curve
  • Decide if this property holds, irrespective of the averaged zeros
  • Find if this property is true for cubic functions with only one or two distinct zeros

  • Graph a cubic function and determine the point where the tangent intersects the curve
  • Divide this region into two parts
  • Find the ratio of the areas of the two regions
  • After the Activity

    Students' answer questions on the Activity sheet.

    Review student results:

  • As a class, discuss questions that appeared to be more challenging
  • Re-teach concepts as necessary