Regents Exam Questions Name: ________________________
F.IF.C.7: Graphing Trigonometric Functions 4
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1
F.IF.C.7: Graphing Trigonometric Functions 4
1 In the interval
0 ≤ x ≤ 2
π
, in how many points will
the graphs of the equations
y = sin x
and
y =
1
2
intersect?
1) 1
2) 2
3) 3
4) 4
2 On the coordinate plane below, sketch at least one
cycle of a cosine function with a midline at
y = −2
,
an amplitude of 3, and a period of
π
2
.
3 On the axes below, graph one cycle of a cosine
function with amplitude 3, period
π
2
, midline
y = −1
, and passing through the point
(0,2)
.
Regents Exam Questions Name: ________________________
F.IF.C.7: Graphing Trigonometric Functions 4
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2
4 A radio wave has an amplitude of 3 and a
wavelength (period) of
π
meters. On the
accompanying grid, using the interval 0 to
2
π
,
draw a possible sine curve for this wave that passes
through the origin.
5 Write an equation for a sine function with an
amplitude of 2 and a period of
π
2
. On the grid
below, sketch the graph of the equation in the
interval 0 to
2
π
.
6 On the graph below, draw at least one complete
cycle of a sine graph passing through point
(0,2)
that has an amplitude of 3, a period of
π
, and a
midline at
y = 2
.
Based on your graph, state an interval in which the
graph is increasing.
Regents Exam Questions Name: ________________________
F.IF.C.7: Graphing Trigonometric Functions 4
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3
7 a) On the axes below, sketch at least one cycle of a
sine curve with an amplitude of 2, a midline at
y = −
3
2
, and a period of
2
π
.
b) Explain any differences between a sketch of
y = 2 sin x −
π
3
−
3
2
and the sketch from part a.
8 Sketch the graph of
y
=
3 sin 2x
in the interval
−
π
≤ x ≤
π
.
9 Sketch and label the function
y = 2 sin
1
2
x
in the
interval
−2
π
≤ x ≤ 2
π
.
Regents Exam Questions Name: ________________________
F.IF.C.7: Graphing Trigonometric Functions 4
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4
10 Graph
t(x)
=
3 sin(2x)
+
2
over the domain
[0, 2
π
]
on the set of axes below.
11 Graph
y = 2 cos
1
2
x
+ 5
on the interval
[0, 2
π
]
,
using the axes below.
12 On the accompanying set of axes, graph the
equations
y = 4 cos x
and
y = 2
in the domain
−
π
≤ x ≤
π
. Express, in terms of
π
, the interval for
which
4 cos x ≥ 2
.
Regents Exam Questions Name: ________________________
F.IF.C.7: Graphing Trigonometric Functions 4
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5
13 a On the accompanying set of axes, sketch the
graph of the equations
y = 2 cos x
in the interval
−
π
≤ x ≤
π
.
b On the same set of axes, reflect the graph drawn
in part a in the x-axis and label it b.
c Write an equation of the graph drawn in part b.
d Using the equation from part c, find the value of
y when
x =
π
6
.
14 On the same set of axes, sketch and label the
graphs of
y = 2 cos
1
2
x
and
y = −1
for the values of
x in the interval
0 ≤ x ≤ 2
π
. State the number of
values of x in the interval
0 ≤ x ≤ 2
π
that satisfy
the equation
2 cos
1
2
x = −1
.
Regents Exam Questions Name: ________________________
F.IF.C.7: Graphing Trigonometric Functions 4
www.jmap.org
6
15 A building’s temperature, T, varies with time of
day, t, during the course of 1 day, as follows:
T = 8 cos t + 78
The air-conditioning operates when
T ≥ 80°F
.
Graph this function for
6 ≤ t < 17
and determine, to
the nearest tenth of an hour, the amount of time in
1 day that the air-conditioning is on in the building.
16 The tide at a boat dock can be modeled by the
equation
y = −2 cos
π
6
t
+ 8
, where t is the
number of hours past noon and y is the height of
the tide, in feet. For how many hours between
t = 0
and
t = 12
is the tide at least 7 feet? [The use of the
grid is optional.]
Regents Exam Questions Name: ________________________
F.IF.C.7: Graphing Trigonometric Functions 4
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7
17 Griffin is riding his bike down the street in
Churchville, N.Y. at a constant speed, when a nail
gets caught in one of his tires. The height of the
nail above the ground, in inches, can be represented
by the trigonometric function
f(t) = −13 cos(0.8
π
t) + 13
, where t represents the
time (in seconds) since the nail first became caught
in the tire. Determine the period of
f(t)
. Interpret
what the period represents in this context. On the
grid below, graph at least one cycle of
f(t)
that
includes the y-intercept of the function.
Does the height of the nail ever reach 30 inches
above the ground? Justify your answer.
18 The average annual snowfall in a certain region is
modeled by the function
S(t) = 20 + 10 cos
π
5
t
,
where S represents the annual snowfall, in inches,
and t represents the number of years since 1970.
What is the minimum annual snowfall, in inches,
for this region? In which years between 1970 and
2000 did the minimum amount of snow fall? [The
use of the grid is optional.]
Regents Exam Questions Name: ________________________
F.IF.C.7: Graphing Trigonometric Functions 4
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8
19 A helicopter, starting at point A on Sunrise
Highway, circles a 2-mile section of the highway in
a counterclockwise direction. If the helicopter is
traveling at a constant speed and it takes
approximately 6.28 minutes to make one complete
revolution to return to point A, sketch a possible
graph of distance (dependent variable) from the
helicopter to the highway, versus time (independent
variable). If the helicopter is north of the highway,
distance (d) is positive; if the helicopter is south of
the highway, distance (d) is negative. (Disregard
the height of the helicopter.) State the equation of
this graph.
20 The ocean tides near Carter Beach follow a
repeating pattern over time, with the amount of
time between each low and high tide remaining
relatively constant. On a certain day, low tide
occurred at 8:30 a.m. and high tide occurred at 3:00
p.m. At high tide, the water level was 12 inches
above the average local sea level; at low tide it was
12 inches below the average local sea level.
Assume that high tide and low tide are the
maximum and minimum water levels each day,
respectively. Write a cosine function of the form
f(t) = A cos(Bt)
, where A and B are real numbers,
that models the water level,
f(t)
, in inches above or
below the average Carter Beach sea level, as a
function of the time measured in t hours since 8:30
a.m. On the grid below, graph one cycle of this
function.
People who fish in Carter Beach know that a
certain species of fish is most plentiful when the
water level is increasing. Explain whether you
would recommend fishing for this species at 7:30
p.m. or 10:30 p.m. using evidence from the given
context.
Regents Exam Questions Name: ________________________
F.IF.C.7: Graphing Trigonometric Functions 4
www.jmap.org
9
21 The resting blood pressure of an adult patient can
be modeled by the function P below, where
P(t)
is
the pressure in millimeters of mercury after time t
in seconds.
P(t) = 24 cos(3
π
t) + 120
On the set of axes below, graph
y = P(t)
over the
domain
0 ≤ t ≤ 2
.
Determine the period of P. Explain what this value
represents in the given context. Normal resting
blood pressure for an adult is 120 over 80. This
means that the blood pressure oscillates between a
maximum of 120 and a minimum of 80. Adults
with high blood pressure (above 140 over 90) and
adults with low blood pressure (below 90 over 60)
may be at risk for health disorders. Classify the
given patient's blood pressure as low, normal, or
high and explain your reasoning.
22 The volume of air in an average lung during
breathing can be modeled by the graph below.
Using the graph, write an equation for
N(t)
, in the
form
N(t) = A s in(Bt) + C
. That same lung, when
engaged in exercise, has a volume that can be
modeled by
E(t) = 2000 sin(
π
t) + 3200
, where
E(t)
is volume in mL and t is time in seconds. Graph at
least one cycle of
E(t)
on the same grid as
N(t)
.
How many times during the 5-second interval will
N(t) = E(t)
?
ID: A
1
F.IF.C.7: Graphing Trigonometric Functions 4
Answer Section
1 ANS: 2 REF: 069522siii
2 ANS:
REF: 082328aii
3 ANS:
REF: 061628aii
4 ANS:
REF: 060832b
ID: A
2
5 ANS:
y = 2 sin 4x
REF: 081934aii
6 ANS:
0 < x <
π
4
REF: 012436aii
7 ANS:
Part a sketch is shifted
π
3
units right.
REF: 081735aii
ID: A
3
8 ANS:
REF: 069040siii
9 ANS:
REF: 019536siii
10 ANS:
REF: 081830aii
ID: A
4
11 ANS:
REF: 062231aii
12 ANS:
−
π
3
≤ x ≤
π
3
.
REF: 080532b
13 ANS:
y = −2 cos x
,
− 3
REF: 069637siii
ID: A
5
14 ANS:
1
REF: 018436siii
15 ANS:
4.2. . . 7.6-6 = 1.6 hours.
13.9-11.3 = 2.6 hours. 1.6+2.6 = 4.2 hours.
REF: 010329b
ID: A
6
16 ANS:
8.
.
REF: 080433b
17 ANS:
pe riod =
2
π
0.8
π
= 2.5
. The wheel rotates once every 2.5 seconds. No, because the maximum
of
f(t) = 26
.
REF: 061937aii
ID: A
7
18 ANS:
10, 1975, 1985, 1995. The minimum of the cosine function is
−1
.
20 + 10(−1) = 10
. .
. . .
REF: 060731b
19 ANS:
d(t) = sin(t)
REF: fall9931b
ID: A
8
20 ANS:
The amplitude, 12, can be interpreted from the situation, since the water level has a
minimum of
−12
and a maximum of 12. The value of A is
−12
since at 8:30 it is low tide. The period of the
function is 13 hours, and is expressed in the function through the parameter B. By experimentation with
technology or using the relation
P =
2
π
B
(where P is the period), it is determined that
B =
2
π
13
.
f(t) = −12 cos
2
π
13
t
In order to answer the question about when to fish, the student must interpret the function and determine which
choice, 7:30 pm or 10:30 pm, is on an increasing interval. Since the function is increasing from
t = 13
to
t = 19.5
(which corresponds to 9:30 pm to 4:00 am), 10:30 is the appropriate choice.
REF: spr1514aii
21 ANS:
The period of P is
2
3
, which means the patient’s blood pressure reaches a high
every
2
3
second and a low every
2
3
second. The patient’s blood pressure is high because 144 over 96 is greater
than 120 over 80.
REF: 011837aii
ID: A
9
22 ANS:
N(t) = 400 sin
2
π
5
t
+ 2400
. 4 times.
REF: 062337aii
