Lesson 14

In this warm-up, students begin to apply their new understandings about graphs to reason about quadratic functions contextually. They evaluate a simple quadratic function, find its maximum, and interpret these values in context. The input values to be evaluated produce an output of 0, reminding students of the meaning of the zeros of a function and their connection to the horizontal intercepts (\(x\)-intercepts) of the graph.

To find the maximum value of the function, students could graph the function, but applying what they learned about the connection between the horizontal intercepts and the vertex (without graphing) would be more efficient. Identify students who make this connection and ask them to share their thinking in the discussion.

Give students access to a calculator.

Select students to share their responses and explanations. Even though students are not asked to graph the function, make sure that they begin to connect the quadratic expression that defines the function to the features of the graph representing that function.

Ask students: “If we graph the equation, what would the graph look like? Where would the intercepts be? Would the graph open up or down? Where would the vertex be?”

Highlight the following points:

• The graph intersects the horizontal axis at \(t=0\) and \(t=0.8\), because \(h(0)\) and \(h(0.8)\) both have a value of 0.
• \(h(0.4)\) is the maximum height of the frog since \(t=0.4\) is halfway between the horizontal intercepts. If we were to graph it, \((0.4,h(0.4))\) would be the vertex.
• The graph opens downward because the coefficient of the squared term is a negative number (-75).

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.

Arrange students in groups of 2. Display the equation \(d = 10 + 406t- 16t^2\) for all to see. Remind students that earlier in the unit, we saw this equation used to model the height of a cannonball that is shot up in the air as a function of time in seconds. The height was measured in feet. Ask students to recall what each term in the equation represents and briefly discuss their thinking.

Next, give students a few minutes of quiet think time to read the first question (all four parts) to themselves and think about their responses. Ask them to share their thoughts with a partner before proceeding to the graphing questions. Clarify if needed that “horizontal intercept” and “vertical intercept” are a more general way to refer to \(x\)- and \(y\)-intercepts when the equation that defines a function uses variables other than \(x\) and \(y\).

Provide access to devices that can run Desmos or other graphing technology. If needed, remind students how to use the available graphing technology to identify the coordinates of points on a graph.

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 \(t\)-coordinate and the other with a negative \(t\)-coordinate. How do we make sense of the negative \(t\)-coordinate in this situation?” (Only the positive one has any meaning in this situation, since negative value of time—the time before the baseball was hit—is not relevant in this model.)
• “How would you estimate the coordinates of the vertex without using a graph? Suppose you know both of the horizontal intercepts.” (We can find the halfway point of the two horizontal intercepts and evaluate the function at that value of \(t\).)

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 (the graph, the numbers in the equation, the descriptions in the questions, etc.), allowing them to reason quantitatively and abstractly (MP2).

As students work, look for students who reason in the ways outlined in the Activity Synthesis so they can share their thinking later.

Give students a moment to read the task statement. Make sure students understand what it means for a baseball to “stay in flight.”

Consider keeping students in groups of 2 and and asking them to think quietly before discussing their thinking with a partner.

Some students may find it challenging to find the zeros of function \(g\) because of the \(\text-16\) in \(\text-16t-1\). Support these students reasoning about the equation \(\text-16t-1=0\) 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 students may face is estimating the location of the vertex. Point out the graph of \(h\) and ask them how the vertex relates to the horizontal intercepts. It is not expected students will compute the precise location by averaging \(\text-\frac{1}{16}\) and 4. Encourage them to use the number between 0 and 4 as a good estimate of the \(t\)-coordinate of the vertex. Encourage struggling students to use technology to graph \(g\) 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 \(h\) intersects the horizontal axis around 5.
  • Function \(g\) is given in factored form, so the zeros will be a very small negative number (about \(\text- \frac{1}{16}\)) and the other will be 4.
• The maximum height of Player A’s baseball (function \(h\)) is shown in the graph. For Player B (function \(g\)), we can approximate the halfway point between the horizontal intercepts, which is around \(t=2\), and then find \(g(2)\).
• The baseball was hit at \(t=0\), so to find the height we would look at the vertical intercept (\(y\)-intercept).
  • For function \(h\), this point is shown on the graph.
  • For function \(g\), it is not immediately apparent because the equation is in factored form. If it was in standard form, the constant term is the vertical intercept. We can either expand the \((\text-16t-1)(t-4)\) and write an equivalent expression in standard form, or find \(g(0)\).

This optional activity gives students an additional opportunity to apply what they learned about key characteristics of a quadratic function 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 need to consider what relevant details are missing.

The info gap structure requires students to make sense of problems by determining what information is necessary, and then to ask for information 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 ask increasingly more precise questions until they get the information they need (MP6).

Here is the text of the cards for reference and planning: 

Tell students they will continue to work with interpreting graphs and equations of a function in terms of a situation involving toy rockets.

Explain the info gap structure, and consider demonstrating the protocol if students are unfamiliar with it.

Arrange students in groups of 2. In each group, distribute a problem card to one student and a data card to the other student. After reviewing students’ work on the first problem, give them the cards for a second problem and instruct partners to switch roles.

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 the Kiran’s rocket reached its highest point?” (The rocket launched at 0 seconds and landed at 1.6 seconds. Those times tell us the \(x\)-intercepts of the graph. The rocket will reach its maximum height at the vertex of the graph, which is halfway between the \(x\)-intercepts.)
• “How did you determine how high the rocket went?” (The maximum height is at \(x=0.8\). If we replace \(x\) in the expression \(\text-16x(x-1.6) \) with 0.8, then \(\text-16(0.8)(0.8-1.6)=10.24\).)
• “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 \(x^2\) term when you write it in standard form.)
• “What does the \(\text-16x^2\) in the equation represent?” (The \(\text-16x^2\) represents the influence of gravity.)
• “If we graphed the equation, what would the point \((0.8, 10.24)\) 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 the parts of the equation.