Answer:
See attachments
Step-by-step explanation:
The period of a sinusoidal graph is the length of one complete cycle along the x-axis, representing the interval required for the function to repeat its pattern.
The given functions are in the form:
[tex]y = A \sin (Bx)[/tex]
[tex]y=A\cos(Bx)[/tex]
where:
- |A| = amplitude (distance from the mid-line to the peak or trough)
- Period = 2π / |B|
[tex]\dotfill[/tex]
For y = 2 sin x, first identify the amplitude and period:
[tex]\textsf{Amplitude} = 2[/tex]
[tex]\textsf{Period} = \dfrac{2\pi}{1}=2\pi[/tex]
In this case, the mid-line of the graph of the function is the x-axis (y = 0) as there is no vertical shift (no 'n' units added to or subtracted from the function). This means that the curve oscillates above and below the x-axis, so the maximum value is y = 2 and the minimum value is y = -2.
A period of 2π signifies that the function completes one full cycle over the interval [0, 2π] radians.
Substitute x = 0 into the function to find the starting point of one cycle:
[tex]y=2\sin(0)=0[/tex]
Plot point (0, 0) as the starting point of the function.
Since the period is 2π, add 2π to the x-coordinate of the starting point to find the end of one period:
[tex](0+2\pi, 0)=(2\pi, 0)[/tex]
Plot point (2π, 0) as the end of one period of the function.
Since the mid-line is y = 0, and the starting and ending points are on the mid-line, the y-coordinate of the midpoint between these points will also be y = 0, so plot point (π, 0).
There will be one peak and one trough within this particular cycle as the starting point and endpoint of the period are on the mid-line. These will occur between the starting point and the midpoint (when x = π/2), and the midpoint and the endpoint (when x = 3π/4).
Substitute x = π/2 into the function to determine if it is a maximum or minimum:
[tex]y=2\sin\left(\dfrac{\pi}{2}\right)=2[/tex]
As point (π/2, 2) is a maximum, then x = 3π/4 must be a minimum point, so (3π/4, -2).
In summary, plot the following points and connect them with smooth curve:
- (0, 0)
- (π/2, 2)
- (π, 0)
- (3π/4, -2)
- (2π, 0)
[tex]\dotfill[/tex]
For y = 4 cos 2x, first identify the amplitude and period:
[tex]\textsf{Amplitude} = 4[/tex]
[tex]\textsf{Period} = \dfrac{2\pi}{2}=\pi[/tex]
In this case, the mid-line of the graph of the function is the x-axis (y = 0) as there is no vertical shift (no 'n' units added to or subtracted from the function). This means that the curve oscillates above and below the x-axis, so the maximum value is y = 4 and the minimum value is y = -4.
A period of π signifies that the function completes one full cycle over the interval [0, π] radians.
Substitute x = 0 into the function to find the starting point of one cycle:
[tex]y=4\cos(2\cdot 0)=4[/tex]
Plot point (0, 4) as the starting point of the function.
Since the period is π, add π to the x-coordinate of the starting point to find the end of one period:
[tex](0+\pi, 4)=(\pi, 4)[/tex]
Plot point (π, 4) as the end of one period of the function.
The starting and ending points are maximums (peaks) as their y-coordinates are y = 4. As the period of a function is the horizontal distance between consecutive peaks (or troughs), there will be one minimum between the two maximum points (0, 4) and (π, 4). Therefore, the minimum point is (π/2, -4).
The midpoint between the maximum (0, 4) and minimum (π/2, -4) and the midpoint between the minimum (π/2, -4) and the maximum (π, 4) will lie on the mid-line, which is y = 0. Therefore, plot points (π/4, 0) and (3π/4, 0).
In summary, plot the following points and connect them with smooth curve:
- (0, 4)
- (π/4, 0)
- (π/2, -4)
- (3π/4, 0)
- (π, 4)
[tex]\dotfill[/tex]
For y = 3 sin ¹/₂x, first identify the amplitude and period:
[tex]\textsf{Amplitude} = 3[/tex]
[tex]\textsf{Period} = \dfrac{2\pi}{\frac12}=4\pi[/tex]
In this case, the mid-line of the graph of the function is the x-axis (y = 0) as there is no vertical shift (no 'n' units added to or subtracted from the function). This means that the curve oscillates above and below the x-axis, so the maximum value is y = 3 and the minimum value is y = -3.
A period of 4π signifies that the function completes one full cycle over the interval [0, 4π] radians.
Follow the steps of the first answer to find the following key points. Plot them and connect them with a smooth curve:
- (0, 0)
- (π, 3)
- (2π, 0)
- (3π, -3)
- (4π, 0)
(Note: As there is a character limit of 5,000 characters, there is not enough room to give a full explanation for answer 3).
