Answer:
Certainly! Let's break down the information and steps to graph the sine function
�
=
3
sin
(
�
)
y=3sin(x).
Amplitude and Period:
Amplitude:
3
3 (this is the vertical stretch or compression of the graph).
Period:
2
�
2π (this is the distance it takes for one complete cycle).
Finding Key Points:
To graph the sine function, it's helpful to find key points in one period (from
0
0 to
2
�
2π).
Key points are often found at critical values such as
0
0,
�
/
2
π/2,
�
π,
3
�
/
2
3π/2, and
2
�
2π.
Calculating y-values:
Plug these x-values into the sine function to get corresponding y-values.
For
�
=
0
x=0:
�
=
3
sin
(
0
)
=
0
y=3sin(0)=0
For
�
=
�
/
2
x=π/2:
�
=
3
sin
(
�
/
2
)
=
3
⋅
1
=
3
y=3sin(π/2)=3⋅1=3
For
�
=
�
x=π:
�
=
3
sin
(
�
)
=
0
y=3sin(π)=0
For
�
=
3
�
/
2
x=3π/2:
�
=
3
sin
(
3
�
/
2
)
=
3
⋅
(
−
1
)
=
−
3
y=3sin(3π/2)=3⋅(−1)=−3
For
�
=
2
�
x=2π:
�
=
3
sin
(
2
�
)
=
0
y=3sin(2π)=0
These values give you points to plot on the graph.
Scale:
The scale your teacher provided,
2
�
/
4
=
�
/
2
2π/4=π/2, is the distance between key points.
This means each major division on the x-axis corresponds to
�
/
2
π/2 radians.
So, mark
�
/
2
π/2,
�
π,
3
�
/
2
3π/2, and
2
�
2π on the x-axis, and use the calculated y-values to plot points.
Graph:
Connect the points smoothly to form the sine wave. Remember, the graph repeats every
2
�
2π due to the periodic nature of the sine function.
This should help you graph the sine function
�
=
3
sin
(
�
)
y=3sin(x) on the plane. If you have specific values you're confused about, feel free to provide them for further clarification.
Step-by-step explanation:
