In previous lessons, students studied the correspondence between translations and reflections of graphs and the algebraic transformations of equations. The purpose of this lesson is for students to understand the effect on a graph of multiplying the output of a function by a scale factor. The situations in this lesson were purposefully chosen to be straightforward in order to help students focus on the effect of this type of scale factor. Students begin by considering a parabolic arch and how to modify a quadratic function whose zeros match the bottom of the arch but whose vertex is too large. Using the horizontal intercepts as focal points, students then decide what scale factor to use to “squash” the graph vertically by modifying the outputs of the function. In the following activity, students use a similar process to fit a radical function to a data set, only now they need to “stretch” the output data, which calls for a scale factor greater than 1.
Students work with descriptions, equations, graphs, and input-output pairs throughout the lesson, helping build fluency translating between the representations (MP2). Students also develop their skills fitting functions to data, helping them in mathematical modeling (MP4). They continue this work in the following lessons where they learn to interpret the effect on the graph of a scale factor on the input to a function.
- Calculate the scale factor $k$ needed to transform the output of a function to fit data.
- Describe the effect of a scale factor on the output of a function.
- Let’s stretch and squash some graphs.
Acquire devices that can run Desmos (recommended) or other graphing technology. It is ideal if each student has their own device. (Desmos is available under Math Tools.)
Optionally, you may find a photo of a local, parabolic arch-shaped landmark to replace the photo in "Arch you glad to see me" and "The Hulme Arch Bridge."
- I can calculate the scale factor needed to transform the output of a function to model data.