# Activities

• • • ##### Subject Area

• Math: Algebra I: Rational Functions

• ##### Author 9-12

60 Minutes

• ##### Device
• TI-89 / TI-89 Titanium

TI Connect™

• ##### Accessories

TI Connectivity Cable

• ##### Report an Issue

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.

• 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