COMPOSE A LOCUS—COMPOSITION, DOMAIN, AND RANGE

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 use Sketchpad’s locus construction to create the range of a function with a restricted domain. They then use this constructed range as a tool to investigate function behavior. For instance, students can change the angle of a rotation or the center of a dilation while observing the effect on the range. In the process, students experience functions as mappings that take a set of input points to a set of output points, and use functions as objects to be combined, edited, and animated.

OBJECTIVES

In this activity students will:

• Create a scale factor and center point to define a dilation function.
• Create an angle measurement and center point to define a rotation function.
• Describe both functions using function notation.
• Compose two functions and label the dependent variable using function notation.
• Describe the relative rate of change of the composed function’s variables.
• Restrict the function’s domain to the border of a polygon.
• Construct the range corresponding to the restricted domain as a locus.
• Animate the independent variable over the restricted domain to verify that the locus construction includes all corresponding values of the dependent variable.
• Modify the scale factor and angle measurement to observe and analyze changes in the mapping from domain to range.
• Animate the scale factor and angle measurement to observe and analyze a continuously changing mapping as the function itself is transformed through a subset of its function family.

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 