Unit 7, Lesson 15
Vertex Form
Algebra 1
15.1
Warm-up
Standards Alignment
Building On
Addressing
Building Toward
HSF-IF.C.8.a
Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context.
Instructional Routines
Notice and WonderMaterials
NoneActivity Narrative
The purpose of this Warm-up is to get students thinking about a new form of quadratic, which will be useful when students explore “vertex form” in a later activity. While students may notice and wonder many things about these functions, a comparison of the forms is the important discussion point.
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.
Launch
Arrange students in groups of 2. Display the 2 sets of equations for all to see. Ask students to think of at least one thing they notice and at least one thing they wonder. Give students 1 minute of quiet think time, and then 1 minute to discuss, with their partner, the things they notice and wonder about.
Activity
NoneWhat do you notice? What do you wonder?
Set 1:
Set 2:
Activity Synthesis
Ask students to share the things they noticed and wondered. Record and display their responses without editing or commentary for all to see. If possible, record the relevant reasoning on or near the equations. Next, ask students, “Is there anything on this list that you are wondering about now?” Encourage students to observe what is on display and to respectfully ask for clarification, point out contradicting information, or voice any disagreement.
If the idea that all three functions in a set represent the same thing, and the new form of the third function in each set does not come up during the conversation, ask students to discuss this idea.
15.2
Activity
Instructional Routines
NoneMaterials
NoneActivity Narrative
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 an expression, given in vertex form, in 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.
Launch
Give students a minute to read the task statement. Before students answer the questions, briefly discuss the worked example to make sure that students can follow the reasoning that illustrates the equivalence of the expressions defining and .
15.3
Activity
Materials
To Gather
Graphing technologyActivity Narrative
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.
Lesson Synthesis
To help students see the connections between the three different forms of quadratic expressions and deepen their understanding of each, consider asking questions such as:
- “The standard form has a constant term: the in . The vertex form also has a constant term: the in . Are the two constant terms visible on the graph in the same way?” (No. The in vertex form gives us the -coordinate of the vertex. The in standard form gives us the -coordinate of the -intercept. That said, changing these parameters has the same effect of moving the graph up or down.)
- “In both the vertex and standard form, the squared term has a coefficient (which could be 1). Does this coefficient affect the graph in similar ways?” (Yes, in both forms, the coefficient influences the direction of the opening of the function’s graph and influences how wide or narrow the opening is.)
Student Lesson Summary
Sometimes the expressions that define quadratic functions are written in vertex form. The function is in vertex form and is shown in this graph.
The vertex form can tell us about the coordinates of the vertex of the graph of a quadratic function. The expression reveals that the -coordinate of the vertex is 3, and the constant term, 4, reveals that the -coordinate of the vertex is 4. Here the vertex represents the minimum value of function , and its graph opens upward.
In general, a quadratic function expressed in vertex form is written as . The vertex of its graph is at . The graph of the quadratic function opens upward when the coefficient, , is positive and opens downward when is negative.