Learners will look for, generate, and make sense of patterns, relationships, and algebraic symbols to represent mathematical models while adopting approaches and solutions in novel situations.

MAT-12.AR.14 Identify zeros of polynomial equations when suitable factorizations are available. Use the zeros to construct a rough graph of the function defined by the polynomial.

MAT-09.AR.10 Solve quadratic equations in one variable by inspection (e.g., for x2 = 49) taking square roots, the quadratic formula, and factoring, as appropriate to the initial form of the equation.

MAT-12.NO.09 Apply the Fundamental Theorem of Algebra to determine the number of zeros for polynomial functions. Find all solutions to a polynomial equation.

MAT-12.AR.07 Create equations and inequalities and use them to solve problems. Include equations arising from linear and quadratic functions and simple rational and exponential functions.

MAT-12.AR.10 Derive the quadratic formula from the form 0 = ax2 + bx + c.

MAT-12.AR.11 Solve quadratic equations with real coefficients that have solutions of the form a + bi and a - bi.

MAT-12.AR.14 Identify zeros of polynomials when suitable factorizations are available. Use the zeros to construct a rough graph of the function defined by the polynomial.

MAT-12.AR.17 Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically.

Graphing Functions

MAT-08.AR.F.05 Describe qualitatively the functional relationship between two quantities by analyzing a graph, including where the function is constant, increasing, or decreasing; linear or nonlinear; and discrete or continuous. Create a graph that exhibits the qualitative features of a function described.

MAT-09.AR.F.01 Determine whether a relationship is a function given a table, graph, or words, identifying x as an element of the domain and f(x) as an element in the range. Determine the domain and range of a function in context.

MAT-09.AR.F.03 Sketch key features (to include intercepts, maximums, minimums, and lines of symmetry, where applicable) of linear, exponential, and quadratic functions modeling the relationship between two quantities using tables, graphs, written descriptions, and equations.

MAT-09.AR.F.09 Identify the effect of transformations on the graph of a linear, absolute value, or quadratic function by replacing f(x) with af(x), f(x - h), and f(x) + k, for specific values of a, h, and k (both positive and negative). Find the value of a, h, and k given the graph of the function.

MAT-09.AR.F.10 Find the inverse of a linear function and describe the relationship between the domain, range, and graph of the function and its inverse. Graph the inverse of a linear function.

MAT-09.AR.F.12 Identify, using graphs or tables, the solution(s) to linear or exponential functions f(x) = g(x) as xvalues that result in equivalent y-values.

MAT-12.AR.14 Identify zeros of polynomials when suitable factorizations are available. Use the zeros to construct a rough graph of the function defined by the polynomial.

MAT-12.AR.16 Identify, using graphs, technology, tables, or successive approximations, that the solution(s) to the equation f(x) = g(x) is the x-value(s) that result in the y-values of f(x) and g(x) being the same.

MAT-12.AR.F.04 Identify the effect of transformations on the graph of a function by replacing f(x) with af(x), f(bx), f(x - h), and f(x) + k, for specific values of a, h, and k (both positive and negative). Find the values of a, b, h, and k given the graph of the function. Recognize even and odd functions from their graphs and equations.

MAT-12.AR.F.08 Use tables, graphs, verbal descriptions, and equations to interpret and sketch the key features of a function modeling the relationship between two quantities.

MAT-12.AR.F.09 Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.

MAT-12.AR.F.10 Graph functions expressed symbolically and show key features of the graph by hand in simple cases and using technology for more complicated cases.

MAT-12.AR.F.11 Analyze and graph functions expressed symbolically (by hand in simple cases and using technology for more complicated cases), identifying key features of the graph.

MAT-12.AR.F.12 Compare the end behavior of linear, quadratic, and exponential functions using graphs and/or tables to show that a quantity increasing exponentially eventually exceeds a quantity increasing as a linear or quadratic function