In this activity, students examine key features of the graph of a quadratic, including intercepts, vertex, and line of symmetry. In the associated Algebra 1 lesson, students begin working with quadratic equations in vertex form. Making graphing technology available gives students an opportunity to choose appropriate tools strategically (MP5).

The purpose of the discussion is to identify ways to find intercepts, vertex, and line of symmetry for a quadratic function that is graphed.

Direct students’ attention to the reference created using Collect and Display. Ask students to share their responses and methods. Invite students to borrow language from the display as needed. As they respond, update the reference to include additional phrases.

Here are some questions for discussion:

• “For this graph, the vertex is on the line of symmetry. Will that always be the case for quadratic equations? Explain your reasoning.” (Yes, the vertex is always on the line of symmetry. If the vertex was not on the line of symmetry, there would be 2 of them, and quadratic equations have only one vertex.)
• “Is the line of symmetry always halfway between the -intercepts? Explain your reasoning.” (As long as there are two -intercepts, the line of symmetry will always be halfway between them. Since the intercepts are along the same horizontal line (the -axis), the vertical line of symmetry should always be halfway between them.)

In this activity, students gather what information they can about quadratic functions without graphing at all and by only looking at the graph. Students must look for and make use of structure (MP7) when they determine information about a quadratic graph without the -axis labeled.

Monitor for students who find the -intercept from the vertex form by:

1. Finding the standard form
2. Substituting 0 for and solving for the -coordinate of the intercept

The purpose of the discussion is to determine methods for finding important points from the equation as well as from the graph. Select students to share their solutions, including previously identified students. Here are some questions for discussion:

• “Is the -intercept easier to find with a function in factored form, like , or in standard form, like ? Which is easier to find the -intercept? Which is easier to find the vertex?” (The -intercepts are easiest to find in the factored form. The -intercept is easiest to find from the standard form. The vertex is easiest to find from the factored form since it is halfway between the -intercepts, which are easiest to find from the factored form.)
• “What could be added to the graph to improve the information you know about the coordinates for the -intercept and vertex?” (A grid or at least marks along the axes would help improve estimates for the coordinates.)