Earlier in this unit, students read off of a diagram the position of an object dropped from the top of a building and then observed that the data is modeled by a quadratic function. This lesson continues to develop this idea with an additional layer of complexity, appropriate for this stage in the unit.

Invite students to share their graph and interpretations of the features of the graph. Discuss with students:

• “The graph shows two horizontal intercepts, one with a positive -coordinate and the other with a negative -coordinate. How do we make sense of the negative -coordinate in this situation?” (Only the positive intercept has any meaning in this situation, because a negative value of time is not relevant in this model.)
• “How would you estimate the coordinates of the vertex from the horizontal intercepts?” (We can find the halfway point of the two horizontal intercepts and evaluate the function at that value of .)

This activity prompts students to analyze two quadratic functions—one represented by a graph, and the other by an equation—and solve a problem. Graphs can make it easier to compare functions, but because students are asked not to graph the second function, they need to rely on their knowledge of the connections between equations and their graphs to make the comparison. To answer the questions, students have to interpret various pieces of the given information, such as the graph, the numbers in the equation, and the descriptions in the questions, which allows them to reason quantitatively and abstractly (MP2).

As students work, look for students who reason in the ways outlined in the Activity Synthesis, and ask them to share their thinking later.

Some students may find it challenging to find the zeros of function because of the  in ( ). Support these students’ reasoning about the equation by asking whether this equation will have a positive or negative solution and whether it will have a solution close to 0 or not. Another challenge that students may face is estimating the location of the vertex. Point out the graph of , and ask them how the vertex relates to the horizontal intercepts. It is not expected that students will compute the precise location by averaging and 4. Encourage them to use the number between 0 and 4 as a good estimate of the -coordinate of the vertex. Encourage struggling students to use technology to graph to check their thinking.

Invite students to share their responses and reasoning. If not mentioned in students’ explanations, highlight the following points:

• To determine which baseball stayed in flight longer, we can compare where the graphs intersect the horizontal axis (or the zeros of the functions), which tell us when the baseball hits the ground.
  • The graph of function  intersects the horizontal axis around 5.
  • Function is given in factored form. One of the zeros will be a very small negative number (about ), and the other will be 4.
• The maximum height of Player A’s baseball (function ) is shown in the graph. For Player B (function ), we can approximate the halfway point between the horizontal intercepts, which is around , and then find .
• The baseball was hit at , so to find the height we would look at the vertical intercept (-intercept).
  • For function , this point is shown on the graph.
  • For function , it is not immediately apparent because the equation is in factored form. If it were in standard form, the constant term would be the vertical intercept. We can either expand the and write an equivalent expression in standard form, or find .

This optional activity gives students an additional opportunity to apply what they learned about key characteristics of a quadratic function and to use that information to solve contextual problems. Unlike in previous activities, where an equation or a graph that represents the function was given up front, here students are presented only with a familiar situation and are asked to consider what relevant details are missing.

The Information Gap structure requires students to make sense of a problem by determining what information is necessary, and then to ask for the information that they need to solve it. This may take several rounds of discussion, if their first requests do not yield the information they need (MP1). It also allows them to refine the language they use and to ask increasingly more precise questions until they get the information that they need (MP6).

After students have completed their work, share the correct answers, and ask students to discuss the process of solving the problems. Here are some questions for discussion:

• “How did you determine when Kiran’s rocket reached its highest point?” (The rocket launched at 0 seconds and landed at 1.6 seconds. Those times tell us the -intercepts of the graph. The rocket will reach its maximum height at the vertex of the graph, which is halfway between the -intercepts.)
• “How did you determine how high the rocket went?” (The maximum height is at . If we replace in the expression  with 0.8, then .)
• “How do you know that the function that models the flight of Kiran’s toy rocket is quadratic?” (The equation that models the flight of the rocket has an term when you write it in standard form.)
• “What does the in the equation represent?” (The represents the influence of gravity.)
• “If we graphed the equation, what would the point represent?” (The vertex of the graph.)

Highlight for students how the information in the problem related to the vertex and intercepts of the graph and to the parts of the equation.