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Algebra 2: Lesson 14-5 Modeling with Trig Functions
Learning Goals:
1) How can we write a trigonometric function that models cyclical behavior?
2) How can we describe the parameters of a trig functions within the context of the problem?
Warm – Up
Determine if each of the following function is cosine function or sine function. Then determine if the
leading coefficient is positive or negative.
Parameters of Trig Functions in a real-world scenario

You almost always want to use a cosine function to model a real-world scenario because it is easier
to locate the maximum point to start the curve than it is to find the point that lies on the midline.




| |
where


Recall the equation of a trig function is always in the form:
(
)
(
)
Example #1
In an amusement park, there is a small Ferris wheel, called a kiddie wheel,
for toddlers. The points on the circle in the diagram to the right represent
the position of the cars on the wheel. The kiddie wheel has four cars,
makes one revolution every minute, and has a diameter of
feet. The
distance from the ground to a car at the lowest point is feet. Assume
corresponds to a time when car 1 is closest to the ground.
Height (feet)
a) Sketch the height function for car 1 with respect to time as the Ferris
wheel rotates for two minutes.
Time (minutes)
b) Find a formula for a function that models the height of car 1 with respect to time as the kiddie wheel
rotates.
Example #2
Distance from
wall (feet)
Once in motion, a pendulum’s distance varies sinusoidally from 12 feet to 2 feet away from a wall every
12 seconds.
a) Sketch the pendulum’s distance D from the wall over a 1-minute interval as a function of time t.
Assume t = 0 corresponds to a time when the pendulum was furthest from the wall.
Time (seconds)
b) Write a sinusoidal function for D, the pendulum’s distance from the wall, as a function of time since
it was furthest from the wall.
Example #3
The tides in a particular bay can be modeled using a sinusoidal function. The maximum depth of water is
36 feet, the minimum depth is 22 feet and high-tide is hit every 12 hours.
( )
Write a cosine function in the form
, where t represents the number of hours since
high-tide and d represents the depth of water in the bay.
Example #4
A Ferris wheel is constructed such that a person gets on the wheel at its lowest point, five feet above the
ground, and reaches its highest point at 130 feet above the ground. The amount of time it takes to
complete one full rotation is equal to 8 minutes. A person’s vertical position, y, can be modeled as a
( )
function of time in minutes since they boarded, t, by the equation
. Sketch a graph of
a person’s vertical position for one cycle and then determine the values of A, B, and C. Show the work
needed to arrive at your answers.