The purpose of this Warm-up is for students to practice extracting information about a context from representations that model the context, which will be useful when students compare situations represented in different ways in a later activity. While students may notice and wonder many things about these representations, connecting key points between the representations are the important discussion points.

As an alternative to the prompt “What do you notice? What do you wonder?” you could simply ask students to write down everything they know about this situation based on the representations of the model.

This activity is an opportunity to connect associated inputs and outputs expressed in function notation with statements about the situation modeled by the function. For example, if someone notices that the arrow is launched from a height of 18 feet, connect this to .

This prompt gives students opportunities to see and make use of structure (MP7). The specific structure they might notice is when important moments in the situation are shown in each representation.

Ask students to share the things they noticed and wondered. Record and display their responses without editing or commentary. If possible, record the relevant reasoning on or near the representations. Next, ask students, “Is there anything on this list that you are wondering about now?” Encourage students to observe what is on display and to respectfully ask for clarification, point out contradicting information, or voice any disagreement.

If the following ideas do not come up during the conversation, demonstrate or ask students to discuss:

• “How can the key points be expressed using function notation? For example, the arrow is launched from a height of 18 feet. How can that be shown using function notation?” ()

• “One of the horizontal intercepts is negative. Why are only positive values of meaningful in this situation?” ( is the time since the arrow was launched. Before the arrow is launched (at negative values of ), the model is not a fair representation of its height.)

• “Where can each of the key points be seen in each representation? For example, from the table it is clear that the arrow hits the ground at 2.25 seconds. How can we see that same information in the graph or equation?” (On the graph is one of the horizontal intercepts. Using the equation, we could solve for the zeros of the function, and one will be .)

• “If you knew only the equation, how could you find the other key points?” (Time permitting, demonstrate how to use graphing technology to create a graph, set an appropriate graphing window, and extract information like the coordinates of the intercepts and the maximum. Also demonstrate how to view a table of inputs and outputs.)

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.

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.