Modelling Periodic Behaviour
- A periodic function is a function which exhibits a pattern that repeats itself regularly
- A periodic pattern can be modelled using a periodic function
- One repetition of a periodic pattern is called a cycle.
- The horizontal length of a cycle on a graph is called the period. The period may be in units of time or other units of measurement
- A function is periodic if there is a positive number, p, such that f (x + p) = f(x). The least value of p that works is the period of the function.
- f(x + np) = f(x), where p is the period and n is any integer.
- The amplitude of a periodic function is half the difference between the maximum value and the minimum value in a cycle.
- The mean value of a periodic function is half the sum between the maximum value and the minimum value in a cycle.
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The Sine Function and the Cosine Function
- Graphs exhibiting the curved form of a sine wave are called sinusoidal functions, regardless if they are written in terms of sinx or cosx
- The graphs of y = sinx and y = cosx both exhibit the same characteristics: both have a period of 360 degrees, both have an amplitude of 1.
- All sinusoidal functions can be graphed using the 5 - point system. The 5 point system relies of the fact that every sinusoidal function will have a maximum, a minimum and 3 zeroes. These 5 points are all equally spaced along the x-axis, therefore to determine how far apart they should be we divide the period by 4.
- The graph of y = sinx can be drawn by beginning at the origin, then increasing the in the positive direction of x until a y-value of 1 is reached, then the graph will decrease.
- The graph of y = cosx can be drawn by beginning at the point (0,1), then decreasing in the positive direction of x until a y-value of 0 is reached, then the graph will increase.
the_sine_function_and_cosine_function.doc | |
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Stretches of Trigonometric Functions
- In the equation y = asinkx and y = acoskx, the parameter a represents a stretch/compression vertically; the parameter k represents a stretch/compression horizontally.
- The a parameter will represents the amplitude of the function
- The k parameter will affect the period of the function. A horizontal stretch will increase the period of the function, while a horizontal compression will shorten the period of the function. To determine the period, given a horizontal stretch/compression, the following formula is used: period = 360/k
- To graph horizontal stretches/compressions use the 5 - point method by dividing the period by 4.
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Combinations of Transformations of y = sinx and y = cosx
trig_graphing_assignment.doc | |
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