High School Math Guide

Modeling Periodic Behavior

Unit 6

MATH CORE STANDARDS III.F.TF.5 : Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline. III.F.TF.2 : Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle. III.F.BF.3 : Identify the effect on the graph of replacing f(x) by f(x) + k, kf(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the values of k given the graphs. III.F.TF.1 : Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle. III.F.IF.4 : For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. III.F.IF.7e : Graph exponential and logarithmic functions, showing intercepts and end behavior; and trigonometric functions, showing period, midline, and amplitude. III.F.TF.3 : Use special triangles to determine geometrically the values of sine, cosine, tangent for . / , . 1 , 2 . , and use the unit circle to express the values of sine, cosine, and tangent for − , + , 2 − in terms of their values for x, where x is any real number.

Critical Background Knowledge

• Transformations of a parent graph (I.F.BF.3, II.F.BF.3) • Knowledge of unit circle trigonometry (III.F.TF.2)

• Understand radian measure (III.F.TF.1) • Define trigonometric ratios (II.G.SRT.6)

• Use trigonometric ratios to solve problems (II.G.SRT.8) • Relate the domain of a function to its graph (II.F.IF.5) • Understanding transformations on functions (I.F.BF.3, II.F.BF.3) • Recognize even and odd functions from a graph and as algebraic expressions (II.F.BF.3) • Graph functions with and without technology (I.F.IF.7, II.F.IF.7) • Use geometric descriptions of rigid motions to transform figures and predict the effect of transformations (I.G.CO.6) • Prove that all circles are similar (II.G.C.1) • Derive the fact that the length of an arc is proportional to the radius (II.G.C.5) • Interpret key features of graphs and tables (I.F.IF.4, II.F.IF.4) • Find the average rate of change on a specified interval (I.F.IF.6, II.F.IF.6)

• Graph functions and identify key features (F.IF.7, II.F.IF.7) • Compare functions using key features (I.F.IF.9, II.F.IF.9) Continued on the next page…