consider y=-2+5cos(20x-5pie)
the period is?
the frequency is?
the vertical shift is?
the horizontal shift is?
more standard form
y = 5 cos 20(x - π/4) - 2
period = 2π/20 = π/10
frequency is 20 cosine curves per 2π radians
vertical shift : 2 units down
horizontal shift: π/4 to the right
To find the period, frequency, vertical shift, and horizontal shift for the given function y = -2 + 5cos(20x - 5π), we can analyze each aspect separately.
1. Period:
The period of a periodic function is the distance between two consecutive peak/trough points or the length of one complete cycle. In the given function, the period can be calculated using the formula:
period = (2π) / |b|,
where b is the coefficient of x inside the cosine function. In this case, b = 20. Therefore, the period is:
period = (2π) / |20| = π / 10.
So, the period of the given function is π / 10.
2. Frequency:
The frequency of a periodic function is the reciprocal of the period. In this case, the frequency is:
frequency = 1 / period = 1 / (π / 10) = 10 / π.
So, the frequency of the given function is 10 / π.
3. Vertical Shift:
The vertical shift is the constant term added or subtracted to the function. In this case, the function is y = -2 + 5cos(20x - 5π), so the vertical shift is -2.
4. Horizontal Shift:
The horizontal shift is determined by the phase shift of the cosine function. In this case, the function is y = -2 + 5cos(20x - 5π). The phase shift is found by setting the argument of cosine equal to zero and solving for x:
20x - 5π = 0
20x = 5π
x = (5π) / 20
x = π / 4.
Therefore, the horizontal shift is π / 4.
Summary:
- The period of the given function is π / 10.
- The frequency of the given function is 10 / π.
- The vertical shift of the given function is -2.
- The horizontal shift of the given function is π / 4.
To identify the period, frequency, vertical shift, and horizontal shift of the given function y = -2 + 5cos(20x - 5π), we need to analyze the equation using the standard form of a cosine function:
y = A * cos(B(x - C)) + D
Where:
A represents the amplitude
B represents the frequency
C represents the horizontal shift
D represents the vertical shift
1. Period: The period of a cosine function is given by the formula T = 2π/B, where B is the coefficient of x in the function's argument. In this case, B = 20, therefore:
T = 2π/20
T = π/10
Hence, the period of the given function is π/10.
2. Frequency: The frequency of a cosine function is the reciprocal of the period. Therefore:
Frequency = 1 / Period
Frequency = 1 / (π/10)
Frequency = 10/π
So, the frequency of the given function is 10/π.
3. Vertical shift: The vertical shift refers to the constant value added or subtracted from the cosine function. In this case, the vertical shift is represented by -2. Therefore, the vertical shift of the given function is -2.
4. Horizontal shift: The horizontal shift represents the phase shift of the function, i.e., the shift in its position on the x-axis. The horizontal shift is determined by the value inside the argument of the cosine function, which is (20x - 5π). To find the horizontal shift, we need to set Bx = C to solve for x. In this case, B = 20 and C = 5π. So:
20x - 5π = 0
20x = 5π
x = 5π/20
x = π/4
Hence, the horizontal shift of the given function is π/4.
Therefore, summarizing the findings:
- The period of the given function y = -2 + 5cos(20x - 5π) is π/10.
- The frequency of the given function is 10/π.
- The vertical shift of the given function is -2.
- The horizontal shift of the given function is π/4.