Project AMP Antonio R. Quesada - Director, Project AMP
Transformations-Translations and Scale Changes
Fluency with transformations of functions is a necessary skill in high school mathematics. These transformations include translations, which shift the coordinate axes along with the graph around the coordinate plane and scale changes, which change the scales of the coordinate system. Changing the scales changes the apparent size and shape of the curve. Both of these types of transformations will be described and illustrated individually and in combination. This lesson will then provide opportunity to gain more of the desired skill.
Key words: transformation, translation, scale change, period, amplitude.
Background Knowledge: Experience with the various parent functions is expected. A working knowledge of transformations; both translations and scale changes. This includes knowing the effect of changing various parameters in the parent functions.
Ohio Academic Content Standards
Geometry and Spatial Sense Standard (GSS)
Students identify, classify, compare and analyze characteristics, properties and relationships of one-, two- and three-dimensional geometric figures and objects. Students use spatial reasoning, properties of geometric objects, and transformations to analyze mathematical situations and solve problems.
Patterns, Functions and Algebra Standard (PFA)
Students use patterns, relations and functions to model, represent and analyze problem situations that involve variable quantities. Students analyze, model and solve problems using various representations such as tables, graphs and equations.
GSS 8-10 F. Represent and model transformations in a coordinate plane and describe the results.
PFA 8-10 C. Translate information from one representation to another.
GSS Gr. 8 #5 Draw the results of translations, reflections, rotations and dilations of objects in the coordinate plane, and determine properties that remain fixed; e.g., lengths of sides remain the same under translations.
GSS Gr. 10 #9 Show and describe the results of combinations of translations, reflections and rotations (compositions); e.g., perform compositions and specify the result of a composition as the outcome of a single motion, when applicable.
PFA Gr. 9 #15 Describe how a change in the value of a constant in a linear or quadratic equation affects the related graphs.
Note: It was really difficult to find benchmarks and grade-level indicators that would match this exactly.
To graph transformed functions by transforming the saxes first and graphing the parent function second:
- Identify and sketch translated axes that go with translated parent functions.
- Identify and sketch changed scales that correspond to functions with scale changes.
- Combine the two transformations in a single problem identifying the order of the two transformations.
Students should work with a graphing calculator to check graph accuracy.
Students could work individually or in pairs, dependent on ability and past experience.
Assessment should include monitoring progress during activities. The objectives for the lesson should also be assessed in both parts of a two-tiered exam. The transformation skills should be tested on the non-calculator portion with “neat” numbers and then using the calculator students should be able to create a quality graph by choosing an appropriate window and justifying their selection.
Note: I’m not really should that these lessons are truly inquiry. We are attempting to guide the students into believing that an alternative approach to graphing transformations of functions is credible.
Files sent to you include:
The Answer Key for the last exercise is not yet available in an electronic form maybe I can E-mail it to you at a later date.