Lesson 15

This warm-up prompts students to analyze two sets of equations that they will study more closely in a later activity. In each set, the three equations define the same function but are written in different forms—factored form, standard form, and vertex form. Noticing and wondering about the features of the equations prepares students to reason later that the expressions defining each output are equivalent.

When students articulate what they notice and wonder, they have an opportunity to attend to precision in the language used to describe what they see (MP6). They might first propose less formal or imprecise language, and then restate their observation with more precise language in order to communicate more clearly.

Display the two sets of equations for all to see. Give students 1 minute of quiet think time and ask them to be prepared to share at least one thing they notice and one thing they wonder. Give students another minute to discuss their observations and questions. Follow with a whole-class discussion.

Ask students to share the things they noticed and wondered. Record and display their responses for all to see. If possible, record the reasoning on or near the relevant equations. After all responses have been recorded without commentary or editing, ask students, “Is there anything on this list that you are wondering about now?” Encourage students to respectfully disagree, ask for clarification, or point out contradicting information, etc. 

This activity introduces students to the vertex form. Students examine the parameters in expressions of this form and the graphs of functions defined by such expressions. They look for structure in the given representations and notice that there are connections between the numbers in each expression and the vertex of the corresponding graph (MP7).

Students also see that we can rewrite the expression in vertex form into another form and show that the expressions are equivalent. Writing equivalent expressions allows students to practice applying the distributive property to expand expressions containing two sums or two differences.

Give students a minute to read the task statement. Before students answer the questions, briefly discuss the worked example to make sure students can follow the reasoning that illustrates the equivalence of the expressions defining \(h\) and \(f\).

Some students may incorrectly think that \((x-3)^2\) is \((x^2-3^2)\). Remind them that \((x-3)^2\) means \((x-3)(x-3)\). Consider pointing out the example in the task statement to help students show that the expressions for \(r\) and \(p\) are equivalent.

Invite students to show that the expressions defining functions \(r\) and \(p\) 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 \(x\)-coordinate of the vertex. The number outside seems to be the \(y\)-coordinate of the vertex.)
• “What doesn’t it tell us?” (It does not allow us to easily see the \(x\)- or \(y\)-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 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 \(a\) and a constant term (MP7). In an upcoming lesson, they will further make sense of how this structure relates to the graph.

Provide access to graphing technology. Consider arranging students in groups of 2. For the first two questions involving graphing, ask one partner to operate the graphing technology and the other to record the group’s observations, and then to switch roles halfway.

Consider pausing for a class discussion after the second question so students can share their observations. When discussing the effects of adding a constant term to \((x-h)^2\), ask students how the vertical movement of the vertex affects the \(x\)-intercepts of the graph. If not mentioned by students, point out that in some cases it produces graphs that are away from the \(x\)-axis and thus have no \(x\)-intercepts.

Encourage students to put their graphing technology out of reach while they work on the third question.

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 \(x\) 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 \(x\) 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 \(y = a(x-h)^2 + k\), the coordinates of the vertex are \((\quad , \quad)\).
• 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 \(y = a(x-h)^2 + k\), the coordinates of the vertex are \((h,k)\). Also point out that when the equation is graphed, the graph of the equation opens upward if \(a\) is positive and opens downward if \(a\) 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, \(y = a(x-h)^2 + k\), produce the behaviors they observed on the graph.