Page 133 - The Ontario Curriculum, Grades 11 and 12: Mathematics, 2007

2.2 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, ampli- tude, period, maximum and minimum values, increasing/decreasing intervals)
Sample problem: Describe and compare the key properties of the graphs of f (x) = sin x and f (x) = cos x. Make some connections between the key properties of the graphs and your understanding of the sine and cosine ratios.
2.3 determine, through investigation using technology, the roles of the parameters d and c in functions of the form y = sin (x – d) + c and y = cos (x – d) + c, and describe these roles in terms of transformations on the graphs of
f(x) = sin x and f(x) = cos x with angles expressed in degrees (i.e., vertical and horizontal translations)
Sample problem: Investigate the graph
f(x) = 2 sin (x – d) + 10 for various values of d, using technology, and describe the effects of changing d in terms of a transformation.
2.4 determine, through investigation using technol- ogy, the roles of the parameters a and k in functions of the form y = a sin kx and
y = a cos kx, and describe these roles in
terms of transformations on the graphs of f(x) = sin x and f(x) = cos x with angles expressed in degrees (i.e., 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) = 2 sin kx for various values of k, using technology, and describe the effects of chang- ing k in terms of transformations.
2.5 determine the amplitude, period, and phase shift of sinusoidal functions whose equations are given in the form f(x) = a sin (k(x – d)) + c or f(x) = a cos (k(x – d)) + c, and sketch graphs of y = a sin (k(x – d)) + c and y=acos(k(x–d))+c byapplyingtransfor- mations to the graphs of f (x) = sin x and
f(x) = cos x
Sample problem: Transform the graph of
f(x) = cos x to sketch g(x) = 3 cos (x + 90°) and h(x) = cos ( 2x) – 1, and state the ampli- tude, period, and phase shift of each function.
2.6 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, using first the sine function and then the cosine function.
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, pressure in sound waves, the height of a tack on a bicycle wheel that is rotating at a fixed speed), through investigation with and with- out 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. Describe how the graph would change if you moved the pendulum further away from the motion sensor. What would you do to generate a graph with a smaller amplitude?
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: The depth, w metres, of water in a lake can be modelled by the func- tion w = 5 sin (31.5n + 63) + 12, where n is the number of months since January 1, 1995. Identify and explain the restrictions on the domain and range of this function.
3.3 pose problems based on applications involv- ing a sinusoidal function, and solve these and other such problems by using a given graph or a graph generated with technology, in degree mode, from a table of values or from its equation
TRIGONOMETRIC FUNCTIONS
3. Solving Problems Involving Sinusoidal Functions
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Mathematics for College Technology
MCT4C