Education Technology

Interesting Properties of Cubic Functions

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