Rationale
The universe if full of phenomenon that are cyclical or follow a repeating pattern. Many of these can be modeled using sinusoidal functions.
Expectations
2.1 describe key properties of periodic functions arising from real-world applications given a numeric or graphical representation
2.2 predict, by extrapolating, the future behavior of a relationship modeled using a numeric or graphical representation of a periodic function
2.3 make connections between the sine ratio and the sine function and between the cosine ratio and the cosine function by graphing the relationship between angles from 0º to 360º and the corresponding sine ratios or cosine ratios, with or without technology, defining this relationship as the function f(x) =sinx or f(x) =cosx, and explaining why the relationship is a function
2.4 sketch the graphs of f(x) =sinx and f(x) =cosx for angle measures expressed in degrees, and determine and describe their key properties
2.5 determine, through investigation using technology, the roles of the parameters a, k, d, and c in functions of the form y =af (k(x – d)) + c, where f(x) =sinx or f(x) =cosx with angles expressed in degrees, and describe these roles in terms of transformations on the graphs of f(x) =sinx and f(x) =cosx
2.6 determine the amplitude, period, phase shift, domain, and range of sinusoidal functions whose equations are given in the form f(x) = asin(k(x – d)) + c or f(x) = acos(k(x – d)) + c
2.7 sketch graphs of y = af (k(x – d)) + c by applying one or more transformations to the graphs of f(x) =sinx and f(x) =cosx, and state the domain and range of the transformed functions
2.8 represent a sinusoidal function with an equation, given its graph or its properties Sample problem: A sinusoidal function has an amplitude of 2 units, a period of 180º, and a maximum at (0, 3). Represent the function with an equation in two different ways.
Settings
Students
The students are expected to have completed Section 1, Representing Functions and Section 2 Solving problems using quadratic of Characteristics of Functions, and so be familiar with the form y=af(k(x-d)) + c, and interpreting the meaning of and applying the transformations represented by a, k, d and c with respect to quadratic functions.
Teacher
The teacher should be familiar with the graphing and data manipulation software available in the school.
Classroom
Ensure LiveMath Viewer is available on the computers.
Assessment
Summative results will be from the Trig-Transform project and the Sunspots paper
Task Overview
- Spaghetti Triganometry Students create graphs of sine and cosine functions using spaghtti and string. - 1 day
- Play with transformations - vocabulary - graph to equation, equation to graph - amplitude, period and phase-shift exercises from Interactive Maths amplitude, period, phase shift activities. - 2 days
- Group-Graph Challenge - go from graphs to equations to properties and back - 2 day
- Trig-Transformers presentation - 3 days
- Sunspots - find amplitude, period of sunspot data - predict next cycle - 4 days