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).

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).