Some students may incorrectly think that is . Remind them that means . Consider pointing out the example in the task statement to help students show that the expressions for  and are equivalent.

Invite students to show that the expressions defining functions  and  are equivalent. Consider pointing out that at the moment it is easier to show equivalence by going from vertex form to standard form than from standard form to vertex form. In a future unit, we will look at how to do the latter.

Then, discuss questions such as:

• “What information does a quadratic expression in the vertex form reveal? How does it show that information?” (It reveals the coordinates of the vertex of the parabola. The number in the parentheses seems to be related to the -coordinate of the vertex. The number outside seems to be the -coordinate of the vertex.)
• “What doesn’t it tell us?” (It does not allow us to easily see the - or -intercepts.)
• “Why do you think this form is used?” (Sometimes we want to know the maximum or the minimum of a function. It is helpful to be able to see it in the expression or equation defining the function.)
• “Can you give an example of a situation when it might be useful to have the relationship modeled using an expression in vertex form?” (Examples: when we are interested in the maximum height of an object in projectile motion, or when we want to know the maximum revenue in a business model.)

Earlier, students noticed that the numbers in a quadratic expression in vertex form are related to the coordinates of the vertex of the graph. Here, they investigate those connections closely. Just as they have done with expressions in standard and factored forms, students use technology to experiment with each parameter of expressions that are in vertex form and study the effects on the graph. In this process, they practice looking for regularity in repeated reasoning (MP8). If working with a partner, students will take turns using the graphing technology and recording observations. As students trade roles explaining their thinking and listening, they have opportunities to explain their reasoning and critique the reasoning of others (MP3).

The work also encourages students to begin looking at the structure of the form—a squared expression with a coefficient and a constant term (MP7). In an upcoming lesson, they will further make sense of how this structure relates to the graph.

Consider displaying the incomplete table and inviting students to write in one of their correct responses. Then ask students what they notice about the structure or composition of the expressions. Draw students’ attention to the following:

• All expressions in the table have a squared term.
• Most of the squared terms contain a sum or a difference of and a number. For the one example where the squared term is not a sum or a difference, we can think of it as having a sum of and 0.
• The squared term may have a coefficient, which can be positive or negative.
• Most expressions have a constant term. For the one example without a constant term, we can think of the constant term as 0.

To help students consolidate their observations, display the following sentence starters, and ask students to complete them based on their work:

• “When a quadratic equation is in vertex form of , the coordinates of the vertex are .”
• “When the equation is graphed, the graph opens upward if . . . .”
• “The graph opens downward if . . . .”

If not mentioned by students, point out that when a quadratic equation is in vertex form of , the coordinates of the vertex are . Also point out that when the equation is graphed, the graph of the equation opens upward if is positive and opens downward if is negative.

Tell students that, in a future lesson, we will take a closer look at how the parts of a quadratic equation in vertex form,  , produce the behaviors that they observed on the graph.