Find the amplitude, period, and phase shift of the following function:
y=-2cos2(θ+45)
Amplitude is 2.
Period is 180 degrees (or one pi or 3.14rad).
Phase shift is 45 degrees to the left.
To find the amplitude, period, and phase shift of the function y = -2cos(2(θ+45)), let's break down the equation step by step.
1. Amplitude:
The general form of a cosine function is y = acos(b(θ-h)) + k, where "a" represents the amplitude. In this case, the coefficient before the cosine term is -2, so the amplitude is |-2| = 2.
2. Period:
To determine the period of the function, we need to find the value of "b" in the general form equation. In this case, "b" is 2, and the period is given by T = (2π)/|b|. So, the period is (2π)/|2| = π.
3. Phase Shift:
The phase shift determines the horizontal shift of the function. It is determined by the term (θ-h) in the general form equation. In this case, we have (θ+45). Since there is a minus sign before the cosine term, we need to represent the phase shift as a shift to the left. To obtain the proper shift, we take the opposite of the value inside the parentheses, which is -45.
Therefore, the amplitude is 2, the period is π, and the phase shift is -45 degrees (or radians).
Note: In the given equation, the parentheses around θ+45 are unnecessary since adding a constant does not alter the values of amplitude, period, or phase shift.