Page 56 - The Ontario Curriculum, Grades 11 and 12: Mathematics, 2007

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3. Solving Problems Involving Sinusoidal Functions
THE ONTARIO CURRICULUM, GRADES 11 AND 12 | Mathematics
2.2 predict, by extrapolating, the future behaviour of a relationship modelled using a numeric or graphical representation of a periodic function (e.g., predicting hours of daylight on a particu- lar date from previous measurements; predict- ing natural gas consumption in Ontario from previous consumption)
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 (e.g., by generating a table of values using a calculator; by unwrapping the unit circle), defining this relationship as the function f(x) =sinx or
f(x) =cosx, and explaining why the relation- ship is a function
2.4 sketch the graphs of f (x) = sin x and
f (x) = cos x for angle measures expressed
in degrees, and determine and describe
their key properties (i.e., cycle, domain, range, intercepts, amplitude, period, maximum
and minimum values, increasing/decreasing intervals)
2.5 determine, through investigation using tech- nology, 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 (i.e., translations; reflections in the axes; vertical and horizontal stretches and compressions to and from the x- and y-axes)
Sample problem: Investigate the graph
f(x) =2sin(x – d) + 10 for various values of d, using technology, and describe the effects of changing d in terms of a transformation.
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
2.7sketchgraphsofy=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
Sample problem: Transform the graph of
f(x) =cos x to sketch g(x) =3cos2x – 1, and state the domain and range of each function.
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.
By the end of this course, students will:
3.1 collect data that can be modelled as a sinu- soidal function (e.g., voltage in an AC circuit, sound waves), through investigation with and without technology, from primary sources, using a variety of tools (e.g., concrete materials, measurement tools such as motion sensors), or from secondary sources (e.g., websites such as Statistics Canada, E-STAT), and graph the data
Sample problem: Measure and record distance−time data for a swinging pendulum, using a motion sensor or other measurement tools, and graph the data.
3.2 identify periodic and sinusoidal functions, including those that arise from real-world applications involving periodic phenomena, given various representations (i.e., tables of values, graphs, equations), and explain any restrictions that the context places on the domain and range
Sample problem: Using data from Statistics Canada, investigate to determine if there was a period of time over which changes in the population of Canadians aged 20–24 could be modelled using a sinusoidal function.
3.3 determine, through investigation, how sinu- soidal functions can be used to model periodic phenomena that do not involve angles
Sample problem: Investigate, using graphing technology in degree mode, and explain how the function h(t) = 5 sin(30(t + 3)) approxi- mately models the relationship between the height and the time of day for a tide with an amplitude of 5 m, if high tide is at midnight.
3.4 predict the effects on a mathematical model (i.e., graph, equation) of an application involving periodic phenomena when the conditions in the application are varied (e.g., varying the conditions, such as speed and direction, when walking in a circle in front of a motion sensor)
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