## SPECIAL EFFECTS—A SWIRLING TRANSFORMATION

Prerequisites:

In Geometric Functions activities students create, manipulate, and experience function concepts by treating geometric transformations as functions with points as their variables.

In this activity students create a new function family, the swirl, which behaves differently from any function students have previously explored. The swirl is easily constructed but surprisingly interesting and complex, prompting students to revisit and refine their concepts of function. (This activity is a lead-in to its companion activity, Animated Special Effects—Swirl a Picture.)

OBJECTIVES

In this activity students will:

• Create a rotation function using a fixed parameter as the angle.
• Redefine the function definition to make the angle of rotation depend on the location of the independent variable.
• Describe the covariation behavior of the modified function.
• Investigate the function as a mapping by restricting its domain, constructing the corresponding range, and comparing their shapes.
• Predict the behavior of other members of the same family, and change the function to test the predictions.
• Animate the function to explore a continuum of members of the function family, and explain the resulting “special effect” in terms of the function that produces it.
• Conduct further investigation to quantify the attributes of the swirl in terms of the relative rates of change of the variables and the shapes of the restricted domain and corresponding range.

COMMON CORE CONNECTIONS
Mathematical Practices

(1) Make sense of problems and persevere in solving them; (2) Reason abstractly and quantitatively; (3) Construct viable arguments and critique the reasoning of others; (4) Model with mathematics; (5) Use appropriate tools strategically; (6) Attend to precision; (7) Look for and make use of structure; (8) Look for and express regularity in repeated reasoning.

Content Standards

8.F1,2; 8.G1; F-IF1,2,9; G-CO2; G-SRT1