Y = 3 sine (one-third x)
1 answer:
Use the form
a
sin
(
b
x
−
c
)
+
d
a
sin
(
b
x
-
c
)
+
d
to find the variables used to find the amplitude, period, phase shift, and vertical shift.
a
=
3
a
=
3
b
=
1
3
b
=
1
3
c
=
0
c
=
0
d
=
0
d
=
0
Find the amplitude
|
a
|
|
a
|
.
Amplitude:
3
3
Find the period using the formula
2
π
|
b
|
2
π
|
b
|
.
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Period:
6
π
6
π
Find the phase shift using the formula
c
b
c
b
.
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Phase Shift:
0
0
Find the vertical shift
d
d
.
Vertical Shift:
0
0
List the properties of the trigonometric function.
Amplitude:
3
3
Period:
6
π
6
π
Phase Shift:
0
0
(
0
0
to the right)
Vertical Shift:
0
0
Select a few points to graph.
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x
f
(
x
)
0
0
3
π
2
3
3
π
0
9
π
2
−
3
6
π
0
x f
(
x
)
0 0
3
π
2
3 3π 0
9
π
2
-3 6π 0
The trig function can be graphed using the amplitude, period, phase shift, vertical shift, and the points.
Amplitude:
3
3
Period:
6
π
6
π
Phase Shift:
0
0
(
0
0
to the right)
Vertical Shift:
0
0
x
f
(
x
)
0
0
3
π
2
3
3
π
0
9
π
2
−
3
6
π
0
x f
(
x
)
0 0
3
π
2
3 3π 0
9
π
2
-3 6π 0
y
=
3
s
i
n
(
1
3
x
)
y
=
3
(
1
3
x
)
a
sin
(
b
x
−
c
)
+
d
a
sin
(
b
x
-
c
)
+
d
to find the variables used to find the amplitude, period, phase shift, and vertical shift.
a
=
3
a
=
3
b
=
1
3
b
=
1
3
c
=
0
c
=
0
d
=
0
d
=
0
Find the amplitude
|
a
|
|
a
|
.
Amplitude:
3
3
Find the period using the formula
2
π
|
b
|
2
π
|
b
|
.
Tap for more steps...
Period:
6
π
6
π
Find the phase shift using the formula
c
b
c
b
.
Tap for more steps...
Phase Shift:
0
0
Find the vertical shift
d
d
.
Vertical Shift:
0
0
List the properties of the trigonometric function.
Amplitude:
3
3
Period:
6
π
6
π
Phase Shift:
0
0
(
0
0
to the right)
Vertical Shift:
0
0
Select a few points to graph.
Tap for more steps...
x
f
(
x
)
0
0
3
π
2
3
3
π
0
9
π
2
−
3
6
π
0
x f
(
x
)
0 0
3
π
2
3 3π 0
9
π
2
-3 6π 0
The trig function can be graphed using the amplitude, period, phase shift, vertical shift, and the points.
Amplitude:
3
3
Period:
6
π
6
π
Phase Shift:
0
0
(
0
0
to the right)
Vertical Shift:
0
0
x
f
(
x
)
0
0
3
π
2
3
3
π
0
9
π
2
−
3
6
π
0
x f
(
x
)
0 0
3
π
2
3 3π 0
9
π
2
-3 6π 0
y
=
3
s
i
n
(
1
3
x
)
y
=
3
(
1
3
x
)
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You can see the three diagram attached. Each link is labeled with the probability: you have probability 1/6 that a six is rolled, and 5/6 that it is not rolled.
To answer the questions, find the path that brings you to the desired outcome, and multiply all the labels you meet.
First question:
To get three sixes, you have to choose the left path at each roll. The probability is always 1/6, so the answer is
Second question:
To get no sixes, you have to choose the right path at each roll. The probability is always 5/6, so the answer is
Third question:
To get exactly one six, it can either be the first, second or third roll.
In all cases, you have to choose the left path once and the right path twice: left-right-right mean that you get the six in the first roll, right-left-right means that you get the six in the second roll, right-right-left means that you get the six in the third roll.
In every case, the left turn has probability 1/6, and the right turn has probability 5/6. The probability of each combination is thus
And since there are three of these combinations, The answer is
Fourth question:
Since the question suggests to use what we already achieved, let's do it: having at least one six is the complementary event of having no sixes at all. If an event has probability p, its complementary has probability 1-p. So, since the probability of no sixes is known, the probability of at least one six is
