(F) Functions Learners will develop a foundational knowledge of functions and use them to model relationships between quantities.

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-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

Transformations

MAT-08.GM.GF.01 Perform single transformations to a figure on or off the coordinate plane and determine whether the figures are congruent or similar.

MAT-08.GM.GF.02 Describe the characteristics of transformations on the coordinate plane using transformation language.

MAT-08.GM.GF.03 Name the type of transformation(s) needed to map a pre-image to its image.

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 f(x) + k, f(x - h) and af(x), for specific values of a, h, and k (both positive and negative).
Find the values of a, h, and k given the graph of the function.

MAT-10.GM.02 Represent transformations in the plane. Describe transformations as functions that take points as outputs. Compare transformations that preserve distance and angle to those that do not (i.e., rigid versus non-rigid motion).

MAT-10.GM.03 Describe the rotations and reflections of a triangle, rectangle, parallelogram, trapezoid, or regular polygon that map each figure onto itself or another figure.

MAT-10.GM.04 Develop or verify the characteristics of rotations, reflections, and translations in angles, circles, perpendicular lines, parallel lines, and line segments.

MAT-10.GM.05 Draw the image of a figure that has undergone a series of transformations [rotation(s),

reflection(s), or translation(s)] of a geometric figure using a variety of methods (e.g., graph paper, tracing paper, or geometry software).

MAT-10.GM.06 Predict the effect of a specified rigid motion on a given figure using geometric descriptions of rigid motions. Determine whether two figures are congruent using the definition of congruence in terms of rigid motions.

MAT-10.GM.14 Verify experimentally and justify the properties of dilations given by a center and a scale factor.

MAT-10.GM.15 Use transformations to decide if two given figures are similar. Apply the meaning of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides.

MAT-10.GM.16 Prove similarity theorems about triangles.

MAT-10.GM.17 Apply knowledge of congruence and similarity criteria for triangles to solve problems and prove relationships in various geometric figures.

MAT-12.AR.F.4 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.