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Section: A1.6.7

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 (A-SSE.A.1$^\star$, F-BF.B.3).
• Use vertex form to write a possible quadratic function given the maximum or minimum of the function or the vertex of its graph (A-SSE.B.3$^\star$, F-IF.C.7a$^\star$).

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.

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1 Building a quadratic function from $f(x) = x^2$

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. They will begin to summarize what they see in the next activity (MP8).

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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$^\star$, F-BF.B.3). The activity can also introduce students informally to the relationship between factored form and the x-intercepts of the graph.