For the given function y=-2cos3(x + 180) - 1. Determine the amplitude, period, phase shift, vertical displacement, domain and range
domain is naturally all reals
vertical shift is -1, so
range is -1-2 to -1+2
amplitude is 2
period is 2pi/3
phase shift is -180
To determine the amplitude, period, phase shift, vertical displacement, domain, and range of the given function y = -2cos(3(x + 180)) - 1, we will break down each aspect separately.
1. Amplitude:
The amplitude of a cosine function is the vertical distance from the middle line (or axis of symmetry) to the highest or lowest point on the graph. In this case, the amplitude is 2 since the coefficient in front of the cosine function is -2. Remember that the negative sign in front of the amplitude only indicates a reflection across the x-axis.
2. Period:
The period of a cosine function is the distance between consecutive peaks (or troughs) on the graph. To find the period, we need to use the formula T = 2π/|B|, where B is the coefficient of x inside the cosine function. In this case, the coefficient is 3. Thus, the period is T = 2π/|3| = 2π/3.
3. Phase Shift:
The phase shift of a cosine function represents the horizontal shift of the graph. To determine the phase shift, we need to isolate the inside of the cosine function and set it equal to zero. In this case, we have 3(x + 180) = 0. Solving for x, we find x = -180. However, since we have x + 180 inside the cosine function, the phase shift is actually -180.
4. Vertical Displacement:
The vertical displacement is the value added or subtracted outside the cosine function, which shifts the entire graph vertically. In this case, the vertical displacement is -1 because of the term -1 at the end of the function.
5. Domain:
The domain of a cosine function is typically all real numbers, unless there are any restrictions due to the presence of other functions or variables. In this case, there are no restrictions, so the domain is (-∞, ∞) or all real numbers.
6. Range:
The range of a cosine function is the set of all possible output values or y-values. Since the amplitude is -2 and the vertical displacement is -1, the range would be [-2 - 1, -2 + 1], which simplifies to [-3, -1].
To summarize:
Amplitude: 2
Period: 2π/3
Phase Shift: -180
Vertical Displacement: -1
Domain: (-∞, ∞)
Range: [-3, -1]