Lesson 17

In this warm-up, students revisit quadratic expressions in different forms and what the expressions reveal about the graphs that represent them. They match equations and graphs representing two quadratic functions. The axes of the graphs are unlabeled, so students need to reason abstractly about the parameters in the expressions and the features of the graphs.

Invite students to share their matches and explanations. If one or more explanations in the Student Response are not mentioned, bring them up so that students can see multiple ways of reasoning about the equations and graphs.

In this activity, students apply what they know about the vertex form to modify a given quadratic expression such that its graph is translated in particular ways to produce a different vertex. Students also consider (by analyzing an argument) whether modifying the squared term of a quadratic expression in standard form produces the same translation on the graph as does modifying the squared term of an expression in vertex form.

As they modify expressions to translate graphs, students make use of structure (MP7). As they analyze an argument about how changing a quadratic expression in standard form affects the graph, they practice explaining their thinking and critiquing the reasoning of others (MP3).

Remind students that they have studied the connections between quadratic expressions and their graphs over several lessons. They have matched expressions and graphs, and sketched graphs of given expressions. Now they will think about how to change one or more parts of a quadratic expression so that its graph has certain features.

Before beginning the activity, ask students to describe the graph representing \(y=x^2\). Make sure students recall that the graph has a vertex at the origin and opens up. If needed, ask them to name some points that the are on the graph, for example, \((\text-3, 9), (\text-2,4), (\text-1, 1), (0,0), (1,1), (2, 4), (3,9)\).

Give students access to graphing technology, but tell students to answer the first question without using technology.

Consider arranging them in groups of 2 and encouraging them to discuss the last question with a partner after some quiet think time. If students use a graph to illustrate that Kiran’s statement is incorrect, encourage them to also use the expression in their justification.

Select students to share their equations for the first question and their explanations for how they knew what modifications to make.

Then, focus the discussion on the third question and how students knew that, when a quadratic expression is in standard form, adding a constant term before squaring the input variable does not translate the graph the same way as when the expression is in vertex form. At this point, students are not expected to come up with a rigorous justification as to why the graph will not translate as Kiran described. They are only to make this observation and consider ways to explain it.

Solicit as many explanations as time permits. If no one mentioned that the expression \(x^2+2x+1\) is not in vertex form and that its parameters do not relate to the graph the same way, bring these points up.

If time permits, consider pointing out that in \(x^2+2x+1\) the input \(x\) shows up in the squared term and the linear term. If we subtract 5 from \(x\) before it is squared but do not subtract 5 from \(x\) before it is multiplied by 2, then the graph does not shift horizontally. (If we graph \((x-5)^2 + 2(x-5) + 1\), the graph does shift 5 units to the right.) In a later course, students will look more closely at the effects on the graph of replacing \(f(x)\) by, for instance, \(f(x) + k\) and \(f(x + k)\).

In this activity, students again apply what they learned about quadratic expressions and their graphs. This time, they solve a contextual problem. Students consider how to change the parameters of a quadratic expression that models the movement of a digitally animated object—a peanut—such that it travels in a parabolic path and meets certain restrictions (MP4).

Providing time to notice and wonder will elicit the idea that this equation shows the horizontal and vertical movement of the peanut, which will be useful when students adjust the equation in a later activity. While students may notice and wonder many things about the graphs and equation, connecting the vertex of the graph to the parameters in the equation are the important discussion points. This warm-up prompts students to make sense of a problem before solving it by familiarizing themselves with a context and the mathematics that might be involved (MP1).

For the second question, there are many possible values of \(h\) and \(k\) that students could use. As students work, notice those who choose to:

• Shift the starting position of the peanut closer to the wall (translating the graph horizontally to the right).
• Stay in the same starting position but shift the vertex to a point closer to the wall.
• Shift both the peanut and the vertex of the parabolic path.

If students choose to use graphing technology to help them experiment with expressions or verify their predictions, they practice choosing tools strategically (MP5).

Some students may randomly choose values for \(h\) and \(k\) until they find a combination whose graph meets the requirements. Encourage them to reason about the problem more systematically, by considering what they learned about the vertex form.

Select students with contrasting strategies to share their equations and explanations for the second question. To verify that students’ proposed equations meet the animation criteria, consider graphing students’ proposed equations for all to see. Highlight the three strategies mentioned in the Activity Narrative.

Point out one distinction between the graphs in this activity and previous graphs that represent projectile motions. In earlier cases, the output represented the height of an object as a function of time, so the graph of the function and physical trajectory of the object may be very different (for example, the object may be going straight up and then fall straight down, but the graph shows a parabola). Here, they see a model where the height of an object is a function of horizontal distance, so the path of the object in a projectile motion does resemble the graph representing the function.

This activity is optional. Use it to allow students to practice writing quadratic expressions to produce several graphs with particular features. Here students will need to know how to restrict the domain of the graph produced using the technology available to them. If students are not already familiar with how to do so, demonstrate how to do so with a simple quadratic function, for example: \(y=x^2\), where the domain is \(\text{-}2 \leq x \leq 2\).

The activity is unlike the ones students have seen before and requires students to transfer and apply their understanding in new ways. It is a chance for students to make sense of a problem and persevere in solving it (MP1).

Provide access to devices that can run Desmos or other graphing technology.