Unit 5
Transformations of Functions
Unit Narrative
In this unit, students consider functions as a whole and understand how they can be transformed to fit the needs of a situation, which is an aspect of modeling with mathematics. Prior to this unit, students have worked with a variety of function types, such as polynomial, radical, and exponential. Students will build on this work in a later unit to transform periodic functions. By saving the introduction of trigonometric functions until after a study of transformations, students have the opportunity to revisit transformations from a new perspective, which reinforces the idea that all functions, even periodic ones, behave the same way with respect to translations, reflections, and scale factors.
The unit begins with students informally describing transformations of graphs, eliciting their prior knowledge, and establishing language that will be refined throughout the unit. Students begin their investigation with vertical and horizontal translations and reflections across the axes. This leads to both algebraic and geometric descriptions of a function as even, odd, or neither.
Students continue to deepen their understanding by exploring the effects of multiplying the input and output of a function by a scale factor. Students then apply their understanding of translations, reflections, and scaling to graphs and equations of many function types. The unit ends with students applying transformations to different functions to model a real-world data set.
Graph of function g on a coordinate plane. X axis from negative 3 to 5. Y axis from negative 4 to 4. From left to right, the function begins around negative 3 comma 2, moves downward and to the right to about negative 1 comma negative 2, stays level until about 2 comma negative 2, moves upward and to the right to about 3 comma 2, moves downward and to the right to 4 comma negative 1, then stays level until 5 comma negative 1.