Lesson 6 - 6: Models of Sinusoidal Functions
Many real-world problems involving periodic behavior can be modeled using sine and cosine functions. Given either a set of data or a graph representing a sinusoidal function, we can determine the equation for the function by calculating the following quantities:
· The amplitude gives the value of “a”.
· The period allows you to calculate “k” .
· The equation of the axis gives the value of “c”.
· The horizontal shift gives the value of “d”.
To determine the horizontal shift, decide which base sinusoidal function you wish to use, y=sinx or y=cosx. Next, determine the horizontal shift needed to translate the base function onto the function modeled by the problem.
Use the a, k, d, and c values to write the equation of the sinusoidal function in the form:
f(x) = asin[k(x – d)] + c OR g(x) = acos[k(x – d)] + c
Ex. 1: The graph below models the depth of the water in an ocean harbour due to the tides as a function of time of day in hours. Represent the function with an equation in two different ways.
Ex. 2: The “SkyWheel” is a Ferris wheel in Niagara Falls. It has a diameter of 53 metres and the ride lasts for 12 minutes for a total of 6 revolutions. Passengers enter the gondola at a height of
2 m above the ground.
a) Determine an equation to model this sinusoidal function.
b) How high is a passenger after 11 minutes and 21 seconds?
Ex. 3: The table below shows the fraction of the moon visible as a function of the day of the year.
a) Determine an equation to model the fraction of moon visible as a function of day of the year.
b) What fraction of the moon is visible on the 100th day of the year?
Homefun: p. 391 #1-3, 6b, 9(omit c), 11
p. 398 #1, 3, 4
CONSOLIDATION:
Sketch the graph of f(x) = 3sin[2x]
6.5 Using Transformations to Sketch the Graphs of Sinusoidal Functions
Unit 6: Sinusoidal Functions
-vertical translation;
-affects eq. of axis, max. & min. values, range
-no effect on period, amplitude or domain ______
-a maximum occurs at 90o ______
-horizontal translation
-no effect on period, amplitude eq. of axis,
domain or range ______
-vertical stretch or compression
-affects max & min, amplitude, and range
-no effect on period or domain ______
-a minimum occurs at 180o ______
-horizontal stretch or compression
-affects period ______
-no effect on amplitude, eq. of axis, max. & min., domain, and range
Combinations of Transformations:
Graphing y = a sin [k(x – d)] + c & y = a cos[k(x – d)] + c
Perform transformations of trigonometric functions in the following order:
1.
2.
3.
State:
1.
2.
3.
4.
5.
Pg, 379 #1-3; Pg 383 #1-2, (3calculators), 4ace, 5, (6-7)ace