In this activity, students create a table consisting of the coordinates of the points around a graph’s vertex. This prepares them to reason using the structure of expressions in the associated Algebra 1 lesson, where they perform a similar analysis without being prompted to create a table first. As students generalize about quadratic equations they are expressing regularity in repeated reasoning (MP8).

The goal of this discussion is to share reasoning about how parts of quadratic equations can adjust the graph.

Direct students’ attention to the reference created using Collect and Display. Ask students to share their reasoning about how to decide if the graphs open up or down. Invite students to borrow language from the display as needed. As they respond, update the reference to include additional phrases.

Display a blank coordinate plane in order to plot points from the table to illustrate the discussion. Here are some questions for discussion:

• “What patterns do you see in the tables?” (The tables show the same value when is 4. The outputs in the tables are the same but opposite.)
• “How can a table help us find the vertex of a graph?” (There are two conditions: it is the greatest or least value of the function, but also the values around it have symmetry.)
• “How can it help us visualize the rest of the graph?” (Comparing the output values on either side of the vertex gives us information about the opening of the parabola.)

The purpose of this discussion is to relate the parameters in the expression to the vertex of the graph. Here are some questions for discussion:

• “How did you decide which table went with each expression?” (I tested points from the table in the expressions. I used the numbers added and subtracted in the expressions to see where the vertex would move.)
• “When a number is added to (inside of the parentheses), how does it affect the vertex of the graph?” (It moves the vertex horizontally.)
• “When a number is added to the end of an expression, how does it affect the vertex of the graph?” (It moves the vertex vertically.)

Point out that the expression defining , for example, could be written , to make it more obvious that the vertex of the graph is . Consider creating a display like this with the parameters labeled, and leaving it displayed in the class for reference.