Describe the transformations that are being applied to the function y = 1−3cos(2x−π).
y = 1−3cos(2x−π)
y = 1−3cos(2(x−π/2))
Assuming your reference function is y = cosx
- you compressed it by a factor of 2, (the period of cosx is 2π, the period of your function is π)
y = cosx ----> y = cos 2x
- you stretched your function vertically by a factor of 3
y = cos 2x ----> y = 3cos 2x
- you reflected it in the x-axis
y = 3cos 2x ----> y = -3cos 2x
- you translated your function up 1 unit
y = -3cos 2x ----> y = 1 - 3cos 2x
- you shifted your function π/2 units to the right
y = 1 - 3cos 2x ---> y = 1 - 3cos 2(x - π/2) OR y = 1 - 3cos (2x - π)
To describe the transformations applied to the function y = 1−3cos(2x−π), we need to analyze the components of the function and how they affect its graph.
1. Vertical shift: The constant term "1" indicates a vertical shift. In this case, the graph is shifted upwards by 1 unit.
2. Amplitude: The coefficient "-3" before the cosine function represents the amplitude. The amplitude of -3 means that the graph's maximum value is 3 units below the horizontal axis, and the minimum value is 3 units above the horizontal axis.
3. Period: The period of the cosine function inside the parentheses, 2x−π, determines the length of one cycle of the graph. The original cosine function has a period of 2π, but we have a 2 in front of the x, which modifies the period. To find the new period, we divide the original period by the coefficient before x. So, the new period is 2π/2 = π.
4. Horizontal shift: The term inside the cosine function, 2x−π, indicates a horizontal shift. To find the value of the shift, we take the argument inside the cosine function and set it equal to zero, then solve for x: 2x−π = 0 ⇔ 2x = π ⇔ x = π/2. This means that the graph is shifted to the right by π/2 units.
In summary, the transformations applied to the function y = 1−3cos(2x−π) are:
- A vertical shift upwards of 1 unit.
- An amplitude of 3 units above and below the horizontal axis.
- A period of π, resulting in one complete cycle of the graph.
- A horizontal shift to the right by π/2 units.