While graphs will vary slightly from one group to the next, lead students to verbalize that it appears that the co-height graph is a horizontal translation of the height graph (and vice versa). Encourage quantitative reasoning by asking students to relate features of the graph back to the scenario of a car rotating around a Ferris wheel. The following questions can guide that discussion. What do the zeros of the graph of the co-height function represent in this situation? They represent the numbers of degrees of rotation where the passenger car is on the vertical line through the center of the wheel, and has a horizontal distance from the center equal to . These are the highest and lowest positions of the car during the ride. What does the vertical intercept of the graph of the co-height function represent in this situation? It represents the radius of the wheel. At the outset of the ride, the car is at the o’clock position, so it has rotated by degrees, and the distance from the center is equal to the radius of the wheel. How are the graphs of the height and co-height functions related to each other? It looks like one graph is a horizontal translation of the other by . Closing (2 minutes) Students should respond to these questions in writing or with a partner. Use this as an opportunity to informally assess their understanding of the height and co-height functions. Closing Why do you think we named the new function the co-height? Both functions measure a distance from the passenger car to one of the axes at various numbers of degrees of rotation of the wheel, so the horizontal measurements are closely related to the vertical measurements. How are the graphs of these two functions alike? How are they different?
While graphs will vary slightly from one group to the next, lead students to verbalize that it appears that the co-height graph is a horizontal translation of the height graph (and vice versa). Encourage quantitative reasoning by asking students to relate features of the graph back to the scenario of a car rotating around a Ferris wheel. The following questions can guide that discussion. What do the zeros of the graph of the co-height function represent in this situation? They represent the numbers of degrees of rotation where the passenger car is on the vertical line through the center of the wheel, and has a horizontal distance from the center equal to . These are the highest and lowest positions of the car during the ride. What does the vertical intercept of the graph of the co-height function represent in this situation? It represents the radius of the wheel. At the outset of the ride, the car is at the o’clock position, so it has rotated by degrees, and the distance from the center is equal to the radius of the wheel. How are the graphs of the height and co-height functions related to each other? It looks like one graph is a horizontal translation of the other by . Closing (2 minutes) Students should respond to these questions in writing or with a partner. Use this as an opportunity to informally assess their understanding of the height and co-height functions. Closing Why do you think we named the new function the co-height? Both functions measure a distance from the passenger car to one of the axes at various numbers of degrees of rotation of the wheel, so the horizontal measurements are closely related to the vertical measurements. How are the graphs of these two functions alike? How are they different?