10.5 Write Trigonometric
Functions and Models
What is a sinusoid?
How do you write a function for a sinusoid?
How do you model a situation with a circular function?
What is sinusoidal regression?
Vocabulary
Sinusoids are graphs of sine and cosine functions.
Use the following equations and find the values of :
𝑎
(amplitude 𝑎 )
2𝜋
𝑏
the period (𝑏 > 0) to find b
ℎ
𝑘
horizontal shift
vertical shift
𝑦 = 𝑎 𝑠𝑖𝑛𝑏 𝑥 − ℎ + 𝑘
𝑦 = 𝑎 cos 𝑏 𝑥 − ℎ + 𝑘
Write a function for the sinusoid shown below.
SOLUTION
STEP 1
STEP 2
Find the maximum value M and minimum
value m. From the graph, M = 5 and m = –1.
Identify the vertical shift, k. The value of k
is the mean of the maximum and
minimum values. The vertical shift is
M + m 5 + (–1)
4
k=
= 2 = 2.
=
2
2
So, k = 2.
STEP 3
Decide whether the graph should be
modeled by a sine or cosine function.
Because the graph crosses the midline
y = 2 on the y-axis, the graph is a sine
curve with no horizontal shift. So, h = 0.
STEP 4
Find the amplitude and period. The
period is π = 2π So, b = 4.
2
b
M – m 5 – (–1)
6
= 2 = 3.
=
The amplitude is a =
2
2
The graph is not a reflection, so a > 0.
Therefore, a = 3.
ANSWER
The function is y = 3 sin 4x + 2.
Jump Rope
Circular Motion
ROPE At a Double Dutch competition, two people
swing jump ropes as shown in the diagram below.
The highest point of the middle of each rope is 75
inches above the ground, and the lowest point is 3
inches. The rope makes 2 revolutions per second.
Write a model for the height h (in feet) of a rope as a
function of the time t (in seconds) if the rope is at its
lowest point when t = 0.
SOLUTION
STEP 1
STEP 2
Find the maximum and minimum values
of the function. A rope’s maximum
height is 75 inches, so M = 75. A rope’s
minimum height is 3 inches, so m = 3.
Identify the vertical shift. The vertical
shift for the model is:
M + m 75 + 3 78
=
=
=
= 39
k
2
2
2
STEP 3
Decide whether the height should be
modeled by a sine or cosine function. When t
= 0, the height is at its minimum. So, use a
cosine function whose graph is a reflection
in the x-axis with no horizontal shift (h = 0).
STEP 4
Find the amplitude and period.
ANSWER
M – m 75 – 3
The amplitude is a =
= 36.
=
2
2
Because the graph is a reflection, a < 0. So, a =
– 36. Because a rope is rotating at a rate of 2
revolutions per second, one revolution is
completed in 0.5 second. So, the period is
2π = 0.5, and b = 4π.
b
A model for the height of a rope is h = – 36 cos 4π t + 39.
Write a function for the sinusoid.
1.
SOLUTION
STEP 1
Find the maximum value M and minimum
value m. From the graph, M = 2 and m = –2.
STEP 2
Identify the vertical shift, k. The value of k
is the mean of the maximum and
minimum values. The vertical shift is
M + m 2 + (–2)
0
k=
= 2 = 0.
=
2
2
So, k = 0.
1.
STEP 3
Decide whether the graph should be
modeled by a sine or cosine function.
Because the graph peaks at y = 2 on the
y-axis, the graph is a cos curve with no
horizontal shift. So, h = 0.
STEP 4
Find the amplitude and period. The
period is 2π= 2π So, b = 3.
3
b
M – m 2 – (–2)
4
= 2 = 2.
=
The amplitude is a =
2
2
The graph is not a reflection, so a > 0.
Therefore, a = 2.
The function is y = 2 cos 3x.
ANSWER
Write a function for the sinusoid.
2.
SOLUTION
STEP 1
Find the maximum value M and minimum
value m. From the graph, M = 1 and m = –3.
STEP 2
Identify the vertical shift, k. The value of k
is the mean of the maximum and
minimum values. The vertical shift is
M + m 1 + (–3)
2
k=
= 2 = 1.
=
2
2
So, k = 1.
2.
STEP 3
Decide whether the graph should be modeled
by a sine or cosine function. Because the
𝟏
graph peaks at y = 1 when x = , the graph is a
𝟐
sin curve with a horizontal shift of 1. So, h = 1.
STEP 4
Find the amplitude and period. The period
is 2 So, b = π.
M – m 1 – (–3)
4
= 2 = 2.
=
The amplitude is a =
2
2
The graph is not a reflection, so a > 0.
Therefore, a = 2.
ANSWER
y = 2 sin π x – 1
Sinusoidal Regression
Sinusoidal regression using a graphing calculator
is another way to model sinusoids.
The advantage is that the graphing calculator
uses all of the data points to find the model.
Energy
The table below shows the number of kilowatt hours K (in
thousands) used each month for a given year by a hangar
at the Cape Canaveral Air Station in Florida. The time t is
measured in months, with t = 1 representing January. Write
a trigonometric model that gives K as a function of t.
SOLUTION
STEP 1
Enter the data in a
graphing calculator.
STEP 2
Make a scatter plot.
STEP 3
Perform a sinusoidal
regression, because
the scatter plot
appears sinusoidal.
STEP 4
Graph the model
and the data in the
same viewing
window.
ANSWER
The model appears to be a good fit. So, a model for
the data is K = 23.9 sin (0.533t – 2.69) + 82.4.
• What is a sinusoid?
Graphs of sine and cosine functions
• How do you write a function for a sinusoid?
Find the values of a, b, h, and k for 𝑦 = 𝑎 𝑠𝑖𝑛𝑏 𝑥 − ℎ + 𝑘
and 𝑦 = 𝑎 cos 𝑏 𝑥 − ℎ + 𝑘
• How do you model a situation with a circular function?
Interpret the characteristics of the data to find the values of
a, b, h, and k and determine whether the graph should be
modeled by a sine function or cosine function.
• What is sinusoidal regression?
Another way to model sinusoids using a graphing calculator.
10.5 Assignment
Page 648, 3-13 odd