In this activity, students determine information about a context from a representation of that context. Making graphing technology available gives students an opportunity to choose appropriate tools strategically (MP5). 

To find the maximum for functions and , the zeros can be averaged to get the -coordinate of the vertex, and then evaluate the function to get the -coordinate. (For example, for , average the zeros -0.25 and 2.5 to get 1.125, and then evaluate .) In fact, due to the symmetry, any two inputs with the same output can be averaged to locate the vertex. This fact is not emphasized in the Algebra 1 lessons, as students will learn to find the vertex of any quadratic function by writing the equation in vertex form in a later unit. So, if all students don’t master this idea at this time, there will be more opportunities later. For , the coordinates of the maximum of the graph can be determined using graphing technology.

The goal of this discussion is to compare how to learn the same key features from different representations. Ensure that students have determined all of the correct information about the contexts from the given representations. Display each representation one at a time, and invite students to share their responses and to explain how they determined the information. Here are some questions for discussion that encourage students to relate specific features to the situation.

• “How did you figure out the launch height?” (I found the height when the time was 0. This is the vertical intercept of the graph.)
• “How did you figure out how long before the object hit the ground?” (I found the positive -intercept which gave me the time when the height was 0.)
• “Which point on the graph tells the maximum height? What do you do if the graph is not given?” (The vertex tells the maximum height. I can see it on the graph. In the table, I knew it was between the two largest outputs. If the graph is not given, I can try to find the halfway point between the zeros, or I can use technology to graph the function.)

This practice activity is similar to the last activity in the associated Algebra 1 lesson, except that students are asked to find specific information about the contexts before being asked to compare them. Students look for and make use of the structure of equations and graphs to complete this work (MP7).

To give students a greater opportunity to communicate their thinking verbally and listen to the reasoning of others, we recommend assigning each student to either work on function or for the first question, and then get together in groups of 2–4 to tackle the remaining questions.

The purpose of this discussion is to compare the representations of the two functions. Much of the conversation will take place within groups. Time permitting, invite students to create a visual display of their solutions and explanations. Give students an opportunity to review other groups’ work, and time to revise and refine their own work. If desired, select a few exemplary visual displays to keep posted in the classroom to support students in later lessons. Here are some questions for discussion:

• “Was it easier to answer the questions using the equation or the graph?” (The graph was easier because the important points were easy to see or already labeled. If they weren’t, it would be harder to know exact answers.)
• “If no points on the graph were labeled, how would you figure out the coordinates of key points on the graph, like the vertex?” (I would approximate to the best of my ability for points that are not exact.)