Activity Overview
In this activity, students use the TI InterActive!™ to construct the expression of a polynomial curve based on its dynamic properties. They explore the behavior of the curve at certain points within its domain. Students also determine the polynomial equations that meet the criteria for the design of a racetrack.

Before the Activity

Install the TI InterActive!™ on the computer
See the attached PDF file for detailed instructions for this activity Print pages 1 - 7 from the attached PDF file for the class

During the Activity
Distribute the pages to the class.
Follow the Activity procedures:
Drawing the existing track:

Draw the semicircle on the left side of the track with t-values ranging from π/2 to 3π/2
Determine the radius of the circle
Enter the parametric equations of the circle
Replace t by t - π and graph the semicircle at the right end
Find transformations for t-values for other sections of the track
Changing the Design of the Track:

Leave the top straight section same, and cut a portion of the bottom one
Reduce the size of the semi-circle
Determine the circumference and the parametric equations of the section
Finding an Expression for the Track:

Determine the point where the track to be constructed meets the small circle
List the values of (x2(t), y2(t) ) when t = π/2
Use a straight edge to draw a straight line joining the two points
Ensure that the entire track looks like a smooth closed curve
Understand that there is sufficient information to solve for 4 parameters
Realize that the polynomial expression will have degree 3 [cubic]
Find linear approximation of the polynomial f(x) = ax^{3} + bx^{2} + cx + d at a given point
Find the equation for the tangent to this polynomial
Graph the line and determine its slope
Solve for the coefficients

After the Activity
Students answer questions on the activity page.
Review student results:

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