The goal of this activity is for students to explore how multiplying the output of a function by a scale factor affects the shape of its graph. Students are also building skills that will help them in mathematical modeling (MP4). While they don’t decide which model to use, students do have an opportunity to transform a function to better fit a given shape.

Monitor for students who identify the scale factor in different ways, such as guess and check or identifying what to multiply the vertex of the parabola by to change from 676 to 25, to share during the whole-class discussion.

The purpose of this discussion is for students to describe how they can adjust a parabolic function in order to model a situation. Ask 2–3 students to share what aspects of the shape they think Han’s function does and does not model well, recording responses for all to see next to the displayed graph. Invite previously selected students to share their responses comparing the height of Han’s graph and the height of the bridge, starting with students who used a guess and check method and ending with students who calculated a scale factor of since .

Conclude the discussion by telling students that the value 0.037 is often called a scale factor. In this instance, the scale factor compressed the graph vertically by a factor of 0.037 toward the -axis. The only points that didn't move were those already on the -axis, which were the two horizontal intercepts. Had the factor been a number greater than 1, we would say that the graph was stretched vertically away from the -axis.

This activity builds on the work done in a previous activity. Students now consider two possible function types for modeling a given data set. They begin by creating a line of best fit for the data and identifying what the linear function does and does not model well. Since the shape of the data shows a downward trend, students then consider a radical function to model the data and use graphing technology to identify an appropriate scale factor to use with a given expression. In this activity, students are building skills that will help them in mathematical modeling (MP4). They don’t decide which model to use, but students have an opportunity to consider two different functions as models and to revise a function to fit a given data set.

If students are unsure how to start identifying an appropriate value for , consider saying:

• “Tell me more about how a scale factor can transform the output of a function.”

• “How could graphing different values of , such as one large, one near 0, and one negative, help you decide on the scale factor?”

The goal of this discussion is for students to understand that if we can recognize the general shape of data (or of an image, as in the previous activity), we can then identify a function type to model the data. Once the function type is known, we can use what we know about graphical transformations to fit a function to the data.

Here are some questions for discussion:

• “What is a reasonable domain for the function ?” (Adult dogs can weigh anywhere from a bit over a pound to almost 350 pounds, so is a reasonable domain.)
• “How did you figure out an appropriate scale factor for ?” (I used the data point . Since , and , I graphed and saw that it was a pretty good fit for the data set.)
• “What other types of functions might work to model this data?” (A quadratic might work with one of the zeros at 0.)