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**Standard(s): **
[MA2015] AL1 (9-12) 3 :

[MA2019] AL1-19 (9-12) 4 :

*Example: Interpret the accrued amount of investment P*(1 + r)^{t} *, where *P* is the principal and *r* is the interest rate, as the product of *P* and a factor depending on time *t*.* [MA2019] AL1-19 (9-12) 5 :

*Example: See *x^{4} - y^{4}* as *(x^{2})^{2} - (y^{2})^{2}*, thus recognizing it as a difference of squares that can be factored as *(x^{2} - y^{2})(x^{2} + y^{2}).

3 ) Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational. [N-RN3]

[MA2019] AL1-19 (9-12) 4 :

4. Interpret linear, quadratic, and exponential expressions in terms of a context by viewing one or more of their parts as a single entity.

5. Use the structure of an expression to identify ways to rewrite it.

Students apply their experiences from Module 4, Topic A as they transform quadratic functions from standard form to vertex form, (*x*) = *a*(*x* - *h*)^{2} + *k* in Topic B. The strategy known as *completing the square* is used to solve quadratic equations when the quadratic expression cannot be factored (A-SSE.B.3b). Students recognize that this form reveals specific features of quadratic functions and their graphs, namely the *minimum* or *maximum of the function* (i.e., the vertex of the graph) and the line of symmetry of the graph (A-APR.B.3, F-IF.B.4, F-IF.C.7a). Students derive the quadratic formula by completing the square for a general quadratic equation in standard form, *y* = *ax*^{2} + *bx* + *c*, and use it to determine the nature and number of solutions for equations when *y* equals zero (A-SSE.A.2, A-REI.B.4). For quadratics with irrational roots, students use the quadratic formula and explore the properties of irrational numbers (N-RN.B.3). With the added technique of completing the square in their toolboxes, students come to see the structure of the equations in their various forms as useful for gaining insight into the features of the graphs of equations (A-SSE.B.3). Students study business applications of quadratic functions as they create quadratic equations and graphs from tables and contexts, and then use them to solve problems involving profit, loss, revenue, cost, etc. (A-CED.A.1, A-CED.A.2, F-IF.B.6, F-IF.C.8a). In addition to applications in business, students solve physics-based problems involving objects in motion. In doing so, students also interpret expressions and parts of expressions in context and recognize when a single entity of an expression is dependent or independent of a given quantity (A-SSE.A.1).