Lesson 8

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. Consider replacing the image with a local parabolic arch-shaped landmark for this warm-up and the activity that follows to increase student interest. While students may notice and wonder many things about these images, the general shape of the arch is the important discussion point.

By engaging with the image of the arch to first become familiar with a context and the mathematics that might be involved, students are making sense of problems (MP1).

Display the image for all to see. Ask students to think of at least one thing they notice and at least one thing they wonder. Give students 1 minute of quiet think time, and then 1 minute to discuss the things they notice with their partner, followed by a whole-class discussion.

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. This activity builds on what students learned previously about the shape of polynomials and the relationship between zeros and factors. 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.

Display the graph with the overlaid axes and grid for all to see throughout the activity. After 5 minutes of work time, pause the class and invite students to share how they labeled the three marked coordinates and what the corresponding points are on the graph of \(H\). Encourage students to use mathematically precise language, such as vertex, zero, and intercept.

Select 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 \(\frac{25}{676} \approx 0.037\) since \(676 \boldcdot \frac{25}{676}=25\).

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 \(x\)-axis. The only points that didn't move were those already on the \(x\)-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 \(x\)-axis.

Building on the work done in the previous activity, students now consider two possible function types for modeling a given data set. Students begin by creating a line of best fit for the data and identify 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.

Provide access to devices that can run Desmos or other graphing technology.

If students are unsure how to start identifying an appropriate value for \(k\), suggest they try graphing a few different values (one large, one close to 0, and one negative, for example) and see which one seems closest, and then deciding how to revise their choice for \(k\) from there.

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 \(F\)?” (Adult dogs can weigh anywhere from a bit over a pound to almost 350 pounds, so \(1 \le x \le 350\) is a reasonable domain.)
• “How did you figure out an appropriate scale factor for \(k\)?” (I used the data point \((60,305)\). Since \(f(60) = 60^{\frac23} \approx 15\), and \(15 \boldcdot 20 = 300\), I graphed \(F(w) = 20w^{\frac23}\) 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.)