The purpose of this activity is to strengthen connections between zeros of a function (input(s) when the output is 0), the output when the input is 0, and the coordinates of the intercept(s) of a graph of the function. Students make sense of problems and persevere in solving them as they determine which inputs are zeros of each function (MP1). When students see connections between the equations and the zeros, they look for and make use of structure (MP7).

The purpose of this discussion is to connect the ideas of zeros, -intercepts, and -intercepts. Display the correct responses to the first question, and ask students to check their work. Invite a student to demonstrate how they know that and (or choose a different function that any students struggled with). The important takeaway is not to look for shortcuts or patterns, but rather to understand by evaluating the function why a given number is a zero of the function.

Invite students to share their responses and reasoning to the second question. Here are some questions for discussion:

• “If you know the equation of a graph, how can you calculate the -intercept?” (The -intercept of a graph corresponds to the output when the input is 0, so to find it, evaluate the function when is 0. For example, , so the coordinates are of an intercept of the graph.)
• “If you know the equation of a graph, how can you calculate the -intercept or intercepts?” (The -intercepts are the inputs that give an output of 0.)
• “What is true of the coordinates of any intercept of any function?” (Any time you are dealing with an intercept, one of the coordinates must be 0. The coordinates of an -intercept are . The coordinates of a -intercept are .)

This activity is an opportunity to practice making the connections discussed in the previous activity. In this activity, 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).

The purpose of this discussion is to clarify connections between equations and graphs with a focus on identifying intercepts. Use Critique, Correct, Clarify to give students an opportunity to improve a sample written response explaining why a graph matches a function, by correcting errors, clarifying meaning, and adding details.

• Display this first draft:
  Function matches with Graph D because function has a negative -intercept and a positive slope.
  Ask, “What parts of this response are unclear, incorrect, or incomplete?” As students respond, annotate the display with 2–3 ideas to indicate the parts of the writing that could use improvement.
• Give students 2–4 minutes to work with a partner to revise the first draft.
• Select 1–2 students or groups to slowly read aloud their draft. Record for all to see as each draft is shared. Then invite the whole class to contribute additional language and edits to make the final draft even more clear and more convincing.

If time allows, invite some students to share the functions they came up with for the last question, and ensure that some quadratic functions are shared. Display these for all to see. Here are some questions for discussion:

• “Which of these functions are quadratic? How do you know?” (The expression that can be written using an term. It has two factors each containing an .)
• “For any of these functions, how can you tell that 7 is a zero of the function?” (It has an output of 0 when the input is 7.)
• “What can you say about the intercepts of the graphs of these functions?” (The graph of each function contains the point  as an -intercept. The -intercept isn’t required to be related to 7, because the -intercept of the graph is the output of the function when the input is 0, rather than when the output is 0.)