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Five-Minute Check (over Lesson 4-3)
Then/Now
New Vocabulary
Key Concept:
Properties of the Sine and Cosine Functions
Key Concept:
Amplitudes of Sine and Cosine Functions
Example 1:
Graph Vertical Dilations of Sinusoidal Functions
Example 2:
Graph Reflections of Sinusoidal Functions
Key Concept:
Example 3:
Graph Horizontal Dilations of Sinusoidal Functions
Key Concept:
Example 4:
Frequency of Sine and Cosine Functions
Real-World Example: Use Frequency to Write a Sinusoidal Function
Key Concept:
Example 5:
Periods of Sine and Cosine Functions
Phase Shift of Sine and Cosine Functions
Graph Horizontal Translations of Sinusoidal Functions
Example 6: Graph Vertical Translations of Sinusoidal Functions
Concept Summary: Graphs of Sinusoidal Functions
Example 7:
Real-World Example: Modeling Data Using a Sinusoidal Function
Over Lesson 4-3
Let (–5, 12) be a point on the terminal side of an
angle θ in standard position. Find the exact
values of the six trigonometric functions of θ.
A.
B.
C.
D.
Over Lesson 4-3
Find the exact value of cot 3π, if defined. If not
defined, write undefined.
A. 0
B. 1
C. –1
D. undefined
Over Lesson 4-3
Find the exact value of csc 210º.
A. –2
B.
C.
D. 2
Over Lesson 4-3
Find the exact value of
A.
B.
C.
D.
.
Over Lesson 4-3
Let
, where cos θ < 0. Find the exact
values of the five remaining trigonometric
functions of θ.
A.
B.
C.
D.
Over Lesson 4-3
Find the exact value of tan
A.
B.
C.
D.
.
You analyzed graphs of functions. (Lesson 1-5)
• Graph transformations of the sine and cosine
functions.
• Use sinusoidal functions to solve problems.
• sinusoid
• amplitude
• frequency
• phase shift
• vertical shift
• midline
Graph Vertical Dilations of Sinusoidal
Functions
Describe how the graphs of f(x) = sin x and
g(x) = 2.5 sin x are related. Then find the amplitude
of g(x), and sketch two periods of both functions
on the same coordinate axes.
The graph of g(x) is the graph of f(x) expanded
vertically. The amplitude of g(x) is |2.5| or 2.5.
Graph Vertical Dilations of Sinusoidal
Functions
Create a table listing the coordinates of the
x-intercepts and extrema for f(x) = sin x for one period
on [0, 2π]. Then use the amplitude of g(x) to find
corresponding points on its graph.
Graph Vertical Dilations of Sinusoidal
Functions
Sketch the curve through the indicated points for each
function. Then repeat the pattern suggested by one
period of each graph to complete a second period on
[2π, 4π]. Extend each curve to the left and right to
indicate that the curve continues in both directions.
Graph Vertical Dilations of Sinusoidal
Functions
Answer: The graph of g(x) is the graph of f(x)
expanded vertically. The amplitude of g(x)
is 2.5.
Describe how the graphs of f(x) = cos x and
g(x) = 5 cos x are related.
A. The graph of g(x) is the graph of f(x)
compressed horizontally.
B. The graph of g(x) is the graph of f(x)
compressed vertically.
C. The graph of g(x) is the graph of f(x) expanded
horizontally.
D. The graph of g(x) is the graph of f(x) expanded
vertically.
Graph Reflections of Sinusoidal Functions
Describe how the graphs of f(x) = cos x and
g(x) = –2 cos x are related. Then find the amplitude
of g(x), and sketch two periods of both functions
on the same coordinate axes.
The graph of g(x) is the graph of f(x) expanded
vertically and then reflected in the x-axis. The
amplitude of g(x) is |–2| or 2.
Graph Reflections of Sinusoidal Functions
Create a table listing the coordinates of key points of
f(x) = cos x for one period on [0, 2π]. Use the amplitude
of g(x) to find corresponding points on the graph of
y = 2 cos x. Then reflect these points in the x-axis to
find corresponding points on the graph of g(x).
Graph Reflections of Sinusoidal Functions
Sketch the curve through the indicated points for each
function. Then repeat the pattern suggested by one
period of each graph to complete a second period on
[2π, 4π]. Extend each curve to the left and right to
indicate that the curve continues in both directions.
Graph Reflections of Sinusoidal Functions
Answer: The graph of g(x) is the graph of f(x)
expanded vertically and then reflected in
the x-axis. The amplitude of g(x) is 2.
Describe how the graphs of f(x) = cos x and
g(x) = –6 cos x are related.
A. The graph of g(x) is the graph of f(x) expanded
horizontally and then reflected in the y-axis.
B. The graph of g(x) is the graph of f(x) expanded
vertically and then reflected in the x-axis.
C. The graph of g(x) is the graph of f(x) expanded
horizontally and then reflected in the x-axis.
D. The graph of g(x) is the graph of f(x) expanded
vertically and then reflected in the y-axis.
Graph Horizontal Dilations of Sinusoidal
Functions
Describe how the graphs of f(x) = cos x and
g(x) = cos
are related. Then find the period of
g(x), and sketch at least one period of both
functions on the same coordinate axes.
Because cos
= cos
, the graph of g(x) is the
graph of f(x) expanded horizontally. The period of g(x)
is
Graph Horizontal Dilations of Sinusoidal
Functions
Because the period of g(x) is 16π, to find
corresponding points on the graph of g(x), change the
x-coordinates of those key points on f(x) so that they
range from 0 to 16π, increasing by increments of
Graph Horizontal Dilations of Sinusoidal
Functions
Sketch the curve through the indicated points for each
function, continuing the patterns to complete one full
cycle of each.
Describe how the graphs of f(x) = sin x and
g(x) = sin 4x are related.
A. The graph of g(x) is the graph of
f(x) expanded vertically.
B. The graph of g(x) is the graph of
f(x) expanded horizontally.
C. The graph of g(x) is the graph of
f(x) compressed vertically.
D. The graph of g(x) is the graph of
f(x) compressed horizontally.
Use Frequency to Write a
Sinusoidal Function
MUSIC A bass tuba can hit a note with a
frequency of 50 cycles per second (50 hertz) and
an amplitude of 0.75. Write an equation for a
cosine function that can be used to model the
initial behavior of the sound wave associated with
the note.
The general form of the equation will be y = a cos bt,
where t is the time in seconds. Because the amplitude
is 0.75, |a| = 0.75. This means that a = ±0.75.
The period is the reciprocal of the frequency or
Use this value to find b.
.
Use Frequency to Write a
Sinusoidal Function
Period formula
period =
|b| = 2π(50) or 100π
Solve for |b|.
Solve for b.
By arbitrarily choosing the positive values of a and b,
one cosine function that models the initial behavior is
y = 0.75 cos 100πt.
Answer: Sample answer: y = 0.75 cos 100πt
MUSIC In the equal tempered scale, F sharp has a
frequency of 740 hertz. Write an equation for a
sine function that can be used to model the initial
behavior of the sound wave associated with
F sharp having an amplitude of 0.2.
A. y = 0.2 sin 1480πt
B. y = 0.2 sin 740πt
C. y = 0.4 sin 370πt
D. y = 0.1 sin 74πt
Graph Horizontal Translations of Sinusoidal
Functions
State the amplitude, period, frequency, and phase
shift of
. Then graph two periods
of the function.
In this function, a = 2, b = 5, and c =
Amplitude: |a| = |2| or 2
Frequency:
Period:
.
Graph Horizontal Translations of Sinusoidal
Functions
Phase shift:
To graph
, consider the graph of
y = 2 sin 5x. The period of this function is
. Create a
table listing the coordinates of key points of
y = 2 sin 5x on the interval
for a phase shift of
, subtract
. To account
from the x-values
of each of the key points for the graph of y = 2 sin 5x.
Graph Horizontal Translations of Sinusoidal
Functions
Sketch the graph of y = 2 sin
through these
points, continuing the pattern to complete two cycles.
Graph Horizontal Translations of Sinusoidal
Functions
Answer: amplitude = 2; period =
frequency =
;
; phase shift =
State the amplitude, period, frequency, and phase
shift of y = 4 cos
A.
amplitude: 4, period:
, frequency:
,phase shift:
B.
amplitude:
, period: 3, frequency:
, phase shift:
C.
amplitude: 4, period: 6π, frequency:
D.
amplitude: –4, period:
, frequency:
, phase shift:
, phase shift:
Graph Vertical Translations of Sinusoidal
Functions
State the amplitude, period, frequency, phase shift,
and vertical shift of y = sin (x + π) + 1. Then graph
two periods of the function.
In this function, a = 1, b = 1, c = π, and d = 1.
Amplitude: |a| = | 1 | or 1
Period:
Frequency:
Phase shift:
Vertical shift: d or 1
Midline: y = d or y = 1
Graph Vertical Translations of Sinusoidal
Functions
Answer: amplitude = 1; period = 2π; frequency =
phase shift = –π; vertical shift = 1
;
State the amplitude, period, frequency, phase shift,
and vertical shift of
.
A.
amplitude: 3, period:
vertical shift: 2
, frequency:
, phase shift:
B.
amplitude: –3, period:
vertical shift: –2
, frequency:
C.
amplitude: 3, period:
vertical shift: 2
, frequency: , phase shift:
,
D.
amplitude: 3, period:
vertical shift: –2
, frequency:
,
, phase shift:
, phase shift:
,
,
Modeling Data Using a
Sinusoidal Function
METEOROLOGY The tides in the Bay of Fundy, in New
Brunswick, Canada, have extreme highs and lows
everyday. The table shows the high tides for one lunar
month. Write a trigonometric function that models the
height of the tides as a function of time x, where x = 1
represents the first day of the month.
Modeling Data Using a
Sinusoidal Function
Step 1
Make a scatter plot of the data and choose
a model.
The graph appears wave-like, so you can
use a sinusoidal function of the form
y = a sin (bx + c) + d or
y = a cos (bx + c) + d to model the data. We
will choose to use y = a cos (bx + c) + d to
model the data.
Modeling Data Using a
Sinusoidal Function
Step 2
Find the maximum M and minimum m
values of the data, and use these values to
find a, b, c, and d.
The maximum and minimum heights are
28.0 and 23.3, respectively. The amplitude a
is half of the distance between the extrema.
a=
The vertical shift d is the average of the
maximum and minimum data values.
Modeling Data Using a
Sinusoidal Function
A sinusoid completes half of a period in the
time it takes to go from its maximum to its
minimum value. One period is twice this
time.
Period = 2(xmax – xmin)
= 2(17 – 10) or 14 xmax = day 17 and
xmin = day 10
Because the period equals
write |b| =
, you can
Therefore, | b | =
Modeling Data Using a
Sinusoidal Function
The maximum data value occurs when
x = 17. Since y = cos x attains its first
maximum when x = 0, we must apply a
phase shift of 17 – 0 or 17 units. Use this
value to find c.
Phase shift formula
Phase shift = 17 and |b| =
Solve for c.
Modeling Data Using a
Sinusoidal Function
Step 3
Write the function using the values for
a, b, c, and d. Use b = .
y = 2.35 cos
is one
model for the height of the tides.
Answer:
TEMPERATURES The table shows the average monthly high
temperatures for Chicago. Write a function that models the
high temperatures using x = 1 to represent January.
A.
B.
C.
D.