The purpose of this Warm-up is to consider the shape of a parabola in the world and recall relevant vocabulary about 2nd degree polynomials, which will be useful when students model the image with a transformed quadratic function in the next activity. While students may notice and wonder many things about these images, the general shape of the arch is the important discussion point.

This Warm-up prompts students to make sense of a problem before solving it by familiarizing themselves with a context and the mathematics that might be involved (MP1).

Ask students to share the things they noticed and wondered. Record and display their responses for all to see. If possible, record the relevant reasoning on or near the image. After all responses have been recorded without commentary or editing, ask students, “Is there anything on this list that you are wondering about now?” Encourage students to respectfully disagree, ask for clarification, or point out contradicting information.

If the parabolic shape of the arch does not come up during the conversation, ask students to discuss this idea along with any relevant vocabulary, such as “vertex” and “horizontal intercepts.”

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.

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.