The purpose of this activity is to clarify a method of rewriting a standard form quadratic equation in vertex form and identifying the vertex.

This activity uses the Worked Example routine. In this routine, students analyze steps used to solve a problem and discuss specific steps with a partner and the whole class. During the discussions, students construct viable arguments to justify the steps taken by the problem solver (MP3). Later, students can apply these strategies to solve similar problems.

The purpose of this discussion is to clarify a method of rewriting a quadratic equation in vertex form and identifying the coordinates of the vertex.

Tell students to open their books or devices. Display the following questions, and give students 1–2 minutes of quiet think time to consider their responses.

• “Why does the solver start by calculating ?” (The solver is using the complete the square method to rewrite the equation in vertex form. To do this, the solver needs to know how much to add and subtract from the equation to make a perfect square binomial.)
• “Why didn’t the solver combine the like terms in Step 2?” (Combining the like terms of would equal -8 and bring the solver back to the starting equation. Keeping the like terms separate, for now, allows the first part of the equation, , to be rewritten as .)
• “How did the solver find the coordinates of the vertex from the vertex form equation?” (Vertex form equations are of the form , where the vertex of the graph is . In this case, and .)

Ask partners to compare their responses to each question and decide if either or both answers are correct. During this partner discussion, monitor to select groups who reason about the questions in different ways. Follow with a whole-class discussion, inviting 1–2 selected groups to share their answers for each question.

If time allows, consider displaying a second worked example next to the first that either:

• Shows a different method of solving a similar problem that students can compare and contrast with the first example (MP7). For example, factoring the equation and finding the -coordinate of the vertex as the halfway point between the -intercepts and using the equation to calculate the -coordinate of the vertex.
• Shows the same method with an error that students can critique and correct (MP3).

In this activity, students convert functions in factored or standard form to vertex form. In the associated Algebra 1 lesson, students use the vertex form to find the maximum or minimum of the quadratic function. This activity supports students by focusing on the mechanics of changing forms and finding the coordinates of the vertex. Students look for and make use of structure (MP7) when they use an equation to identify the vertex of a graph.

In this activity, students compare the values of a function with different -coordinates using equations and graphs. In the associated Algebra 1 lesson, students use the vertex form of a quadratic equation to find the maximum or minimum. This work supports students by focusing on the values of the equation and determining which is greater.

Students critique a statement or response that is intentionally unclear, incorrect, or incomplete and improve it by clarifying meaning, correcting errors, and adding details (MP3).