» One period = the intercepts, the maximum
points, and the minimum points
» Amplitude: of y = a sinx and y = a cosx represent
half the distance between the maximum and
minimum values of the function and is given by
˃ Amplitude = IaI
» Period: Let be be a positive real number. The
period of y = a sinbx and y = a cosbx is given by
˃ Period = (2π)/b
» Sketch the graph y = 2sinx by hand on the
interval [-π, 4π]
» Sketch the graph of y = cos (x/2) by hand over
the interval [-4π, 4π]
» The constant c in the general equations
˃ y = a sin(bx – c)
and
y = a cos(bx – c)
create horizontal translations of the basic sine and
cosine curves.
» One cycle of the period starts at bx – c = 0 and
ends at bx – c = 2π
» The number c/b is called a phase shift
» Sketch the graph y = ½ sin (x – π/3)
» Sketch the graph y = 2 + 3 cos2x
» Find the amplitude, period, and phase shift for
the sine function whose graph is shown. Then
write the equation of the graph.
» For a person at rest, the velocity v (in liters per second)
of air flow during a respiratory cycle (the time from the
beginning of one breath to the beginning of the next) is
t
given by v  .85 sin where t is the time (in seconds) .
3
(inhalation occurs when v> 0 and exhalation occurs
when v < 0 )
a) Graph on the calculator
b) Find the time for one full respiratory cycle
c) Find the number of cycles per minute
d) The model is for a person at rest. How might the
model change for a person who is exercising?
» A company that produces snowboards, which
are seasonal products, forecast monthly sales
t
for 1 year to be S  74.50  43.75 cos 6 where S is the
sales in thousands of units and t is the time in
months, with t = 1 corresponding to January.
a) Graph the function for a 1 year period
b) What months have maximum sales and which
months have minimum sales.
» Throughout the day, the depth of the water at the
end of a dock in Bangor, Washington varies with
the tides. The table shows the depths (in feet) at
various times during the morning.
Time
12am
Depth, y 3.1
2 am
4 am
6 am
8 am
10am
12pm
7.8
11.3
10.9
6.6
1.7
.9
» Use a trigonometric function to model this data.
» A boat needs at least 10 feet of water to moor at
the dock. During what times in the evening can it
safely dock?