Express Quadratic Functions in Vertex Form
• Understand how the structure of vertex form is related to the maximum or minimum value of the function and to the vertex of its graph. • Use vertex form to write a possible quadratic function given the maximum or minimum of the function or the vertex of its graph.
The standard form of a quadratic is not always the most useful form for a given situation. When modeling the path of a projectile, for example, it may be useful to express a function in vertex form in order to find the maximum height. In this section students interpret and construct functions expressed in vertex form. It is possible, but not necessary, that students begin work with converting a function from standard form to vertex form by completing the square. They are not expected to gain fluency in this operation until the next unit.
WHAT: In this exploration students sketch transformations of the graph of a quadratic function given expressions for the transformations in function notation. The task can be done by hand or using Desmos or another graphing calculator.
WHY: The purpose of this activity is for students to explore transforming graphs and seeing how the structure of an expression in vertex form is related to the position and shape of its graph A-SSE.A.1, F-BF.B.3. They will begin to summarize what they see in the next activity MP.8.
WHAT: Students are given graphs of carefully chosen quadratic functions (a few with vertex at the origin, one with the parent function shifted down) and find find equations for the functions with the help of graphing technology MP.5. After students have time to tinker and conjecture about how to modify an equation to shift the graph, the teacher brings the class together to summarize and generalize their findings. Finally, students are given descriptions of several parabolas to practice creating equations and graphs.
WHY: The purpose of this task is to build understanding of how changes to the expression for a function affect its graph by having students come up with equations on their own. A lesson built on this task extends the exploration started in the previous activity. When students summarize their findings, they arrive at the vertex form of a quadratic function F-BF.B.3, F-IF.C.7a, MP.8.
WHAT: Des-Man provides a sandbox for students to play with creating graphs, with the goal of making a face. The interface allows the teacher to create a class and monitor what all students are working on at once, sharing various faces throughout the work together. This could promote collaboration and the students could seek new ideas for their graphs from one another.
WHY: This is a low-risk, rewarding way for students to practice and solidify the generalizations they have made in this section A-SSE.B.3, F-BF.B.3. The activity can also introduce students informally to the relationship between factored form and the x-intercepts of the graph.