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).

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 is not in vertex form and that its parameters do not relate to the graph the same way, bring up these points.

If time permits, consider pointing out that in the input shows up in the squared term and the linear term. If we subtract 5 from before it is squared but do not subtract 5 from before it is multiplied by 2, then the graph does not shift horizontally. (If we graph , 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 by, for instance, and .

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).

For the second question, there are many possible values of and 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 to verify their predictions, they practice choosing tools strategically (MP5).

In the digital version of the activity, students use an applet to see, in use, the equation that they select. The applet allows students to play with different values more freely and to get immediate feedback on their choices. Use the digital version if students need help visualizing the effect of their choices.

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 use the technology available to them in order to restrict the domain of the graph produced. If students are not already familiar with how to do so, demonstrate how to do so with a simple quadratic function, for example: , where the domain is .

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).

Some students may randomly choose values for and 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.