Prior to this course, students learned that an object traveling at a constant speed can be described with a linear function whose graph is a straight line. Here they see a model that accounts for the fact that an object that is launched straight up at a constant speed does not keep going at the same rate when the influence of gravity is taken into account. Adding a quadratic term to a linear function has the effect of curving the graph, because the output values are no longer changing at a constant rate.

If students are unsure how to write an equation to represent the values in the table, ask them to compare how the actual heights of the t-shirt at each second differ from those in the no-gravity case, as shown in the table in the Warm-Up. Finding the differences between the two outputs at the same input values should help students think of the numbers and the functions they saw in the previous lesson.

To generalize the relationship between time and distance, students reason repeatedly with numerical values and look for regularity (MP8). If students opt to use spreadsheet or graphing technology, they practice choosing appropriate tools strategically (MP5).

When comparing the tables, some students may make observations that lack the detail needed to write an equation for the actual height. Prompt them rewrite the outputs for the actual height in terms of the hypothetical height ( , , , and so on).  Show them values of  from a previous lesson to help them see and extend the pattern to write the equation. 

The purpose of this discussion is to describe the effect of the force of gravity on the equation that models a situation. Invite students to share their observations about the two tables (the one from the Warm-Up and the one here) and how the two graphs compare. Highlight responses that suggest that the values in the second table account for the effect of gravity.

Help students see how the output for each value varies across the two tables. When is 1, the output in feet in the second table is 16 less than in the first table. When is 2, there is a difference of 64 feet. When is 3, that difference is 144 feet, and so on. The values 16, 64, 144, . . . correspond to the expression that we saw in the previous lesson (the distance that an object falls in feet, as a function of time in seconds), so we can represent the values in the second table with the equation . Ask students:

• “What do the 5, , and mean in this situation?” (The 5 is the vertical position of the t-shirt before it was launched: 5 feet above ground. In , the 90 tells us the vertical velocity at which it was launched upward into the air. The accounts for the effect of gravity on the height of the t-shirt after it was launched upward.)
• “Why do you think the graph that represents changes from a straight line to a curve when is added to the equation?” (Before that term was added, the height increased by 90 feet every second. Adding decreases how much the t-shirt travels upward by some amount, but that amount gets larger each successive second. Eventually the t-shirt stops increasing in height and starts to fall.)

In this activity, students explore another model of a projectile motion. They graph and interpret a quadratic function in context and begin considering a reasonable domain for the function. Along the way, they practice reasoning concretely and abstractly (MP2). By the end of the lesson, they relate the vertex of the graph to the maximum height of the cannonball and the positive zero of the function to the time when the cannonball hits the ground.

Invite students to share their observations and interpretations of the graph. Highlight these points:

• The graph representing a quadratic function is a parabola, which is a special kind of U shape. We will learn more about the geometry of this shape in a later course, but for now, notice that there is a point at which the graph changes direction, from going up as  increases to going down (or changes from going down to going up). We call this point the vertex of the graph.
• In the previous activity, we plotted a limited set of points, so we could not tell where the vertex of the graph was. In this graph, we are able to identify the vertex. In this situation, the vertex tells us the maximum height that the cannonball reaches and the number of seconds after launch that it takes before it starts to fall.
• In this graph, we can also see that the height of the cannonball is 0 when  is a little less than 20. That point, the horizontal intercept, relates to the zero of the function, or an input value that produces 0 for the output. In this situation, the zero tells us when the cannonball hits the ground.
• Although they are related, the zero of a function is not the same as the horizontal intercept. A zero is a single value for a function that produces an output of 0. A horizontal intercept is a point on a graph represented by a coordinate pair.
• Even though we can continue the graph beyond , in this situation any output values beyond that point would not have any meaning. After the cannonball hits the ground, the function is no longer appropriate for modeling the movement of the cannonball. Likewise, the function is not appropriate before or before the cannon is fired. In this situation, a domain between 0 and just below 20 seconds is appropriate.