Work with the three basic forms
• Understand how the structure of factored form is related to the zeros of a function and the x-intercepts of its graph (A-SSE.A.1$^\star$, F-IF.C.7a$^\star$).
• Select the best form for expressing a quadratic function to illuminate specific features of graphs (A-SSE.B.3$^\star$, F-IF.C.8).
In this section, students write quadratic functions in different forms to illuminate different features of the function. Through the use of graphing technology or by hand, they explore ways to sketch quadratic functions given key features of the graph. Factored form, which has not been explicitly discussed before now, comes into play in these activities and students explore its equivalence to other forms through graphing. They explore expressions in vertex, factored, or standard form and interpret those expressions in terms of a model. By the end of this section students should be fluent with converting from factored or vertex form to standard form. They are also in the process of learning how to construct factored and vertex form from standard form, and become fluent with this in the next unit.
WHAT: Students graph several functions expressed in standard, factored, and vertex form and make observations about equivalence, intercepts, and vertices. They reason through the construction of equations for functions given features their graphs.
WHY: The purpose of this task is to start the process of bringing some order to the study of different forms of quadratic expression, with the eventual goal of knowing the three basic forms and how to transform between them. Using graphs reminds students of the fundamental principle behind equivalence of expressions: expressions are equivalent if they define the same function and therefore have the same graph (MP5).
WHAT: Students are presented with three equivalent forms of an expression representing profit in terms of price, and are asked to decide which form is best suited to answering particular questions about the context.
WHY: The purpose of this task is to help students see structure in expressions and relate that structure to a purpose. The previous task asked, “which expressions are equivalent,” whereas Profit of a Company says “These expressions are equivalent–which one is best for answering different questions about the context”? This task draws important connections between abstract representations and the real relationships they represent (MP4, MP7).
WHAT: In this task, students are given a graph and choose among functions expressed in different forms the ones that could have that graph. It could be used as group exploration or individual practice at this point in the unit.
WHY: The purpose of this task is to further develop students’ ability to see structure in the expression for a function and related it to features of the graph. Students might initially graph the expressions using technology, but it is important that they grapple with what graphs look like without the black box of technology. They should eventually come to see that from the structure of quadratic expressions they can quickly sketch a graph. This task could be opened up into a matching game where students create a sketch of a graph and the rest of the class or their table has to write as many equations as they can that might be represented by that graph (MP7).
WHAT: In this task, students are asked to match given graphs with given equations in all three forms, and explain the reasoning behind their matches.
WHY: Forming quadratics is an activity where the students practice and dive deeper with three different forms of a quadratic function. It could be a formative assessment administered either individually or in groups. Students must notice the equivalence of three forms (because they make the same graph) and the usefulness of one over the other (A-SSE.B.3$^\star$). In this lesson, key pieces of vocabulary, such as roots, zeroes, maximum, minimum, vertex, and axis of symmetry, should be finalized (they have probably came up in earlier tasks) (F-IF.C.8, MP1, MP2, MP3).