Page 68 - The Ontario Curriculum, Grades 11 and 12: Mathematics, 2007

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3. Solving Problems Involving Sine Functions
THE ONTARIO CURRICULUM, GRADES 11 AND 12 | Mathematics
2.4 sketch the graph of f(x) = sinx for angle measures expressed in degrees, and determine and describe its key properties (i.e., cycle, domain, range, intercepts, amplitude, period, maximum and minimum values, increasing/ decreasing intervals)
2.5 make connections, through investigation with technology, between changes in a real-world situation that can be modelled using a periodic function and transformations of the correspon- ding graph (e.g., investigate the connection between variables for a swimmer swimming lengths of a pool and transformations of the graph of distance from the starting point versus time)
Sample problem: Generate the graph of a periodic function by walking a circle of
2-m diameter in front of a motion sensor. Describe how the following changes in the motion change the graph: starting at a differ- ent point on the circle; starting a greater distance from the motion sensor; changing direction; increasing the radius of the circle.
2.6 determine, through investigation using technology, the roles of the parameters a, c, and d in functions in the form f(x) = a sinx, f(x) = sinx + c, and f(x) = sin(x – d), and describe these roles in terms of transformations on the graph of f(x) = sinx with angles expressed in degrees (i.e., translations; reflections in the x-axis; vertical stretches and compressions to and from the x-axis)
2.7 sketch graphs of f(x) = a sinx, f(x) = sinx + c, and f(x) = sin(x – d) by applying transforma- tions to the graph of f(x) = sinx, and state
the domain and range of the transformed functions
Sample problem: Transform the graph
of f(x) = sinx to sketch the graphs of g(x) =–2sinx and h(x) =sin(x –180°), and state the domain and range of each function.
By the end of this course, students will:
3.1 collect data that can be modelled as a sine 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
3.3 pose problems based on applications involving a sine function, and solve these and other such problems by using a given graph or a graph generated with technology from a table of values or from its equation
Sample problem: The height above the ground of a rider on a Ferris wheel can be modelled by the sine function h(x) = 25 sin(x – 90˚) + 27, where h(x) is the height, in metres, and x is the angle, in degrees, that the radius from the centre of the ferris wheel to the rider makes with the horizontal. Graph the function, using graphing technology in degree mode, and determine the maximum and minimum heights of the rider and the measures of the angle when the height of the rider is 40 m.
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