Given y=2-4cos(3x-pi/4), find each of the following, giving the general representation of the location of all minimums. Also provide the graph of two full cycles, labeling everything.
Domain =
Range =
Amplitude =
Period =
Ph. Shft. =
Interval for one
Complete cycle =
Horizontal:
Minimums at
Maximums at
Graph (2 full cycles)
To find the domain of the function y = 2 - 4cos(3x - π/4), we need to determine the values that x can take.
The cosine function is defined for all real numbers, so there are no restrictions on x. Therefore, the domain of the function is (-∞, ∞).
To find the range, we need to determine the values that y can take. The cosine function has a range between -1 and 1, and it is multiplied by -4 in this case. Thus, the range of the function is (-∞, -4] (where the square bracket indicates that -4 is included).
The amplitude of the function is the absolute value of the coefficient of the cosine function, which is 4. Therefore, the amplitude is 4.
The period of the function is given by 2π/|b|, where b is the coefficient of x in the cosine function. In this case, the coefficient is 3, so the period is 2π/3.
The phase shift of the function is given by c/|b|, where c is the constant term inside the cosine function. In this case, the constant term is -π/4, so the phase shift is (-π/4)/3.
The interval for one complete cycle is equal to the period of the function, which is 2π/3.
The horizontal axis (x-axis) is the axis of the cosine function. In this case, it is shifted by -π/4 units to the right (positive direction) due to the phase shift.
To find the locations of the minimums and maximums, we need to consider the amplitude and the vertical shift of the function. The general representation of a cosine function with these parameters can be written as:
y = A*cos(b(x - c)) + D,
where A is the amplitude, b is the coefficient of x, c is the phase shift, and D is the vertical shift.
In this case, A = 4, b = 3, c = -π/4, and D = 2.
The minimums occur when the cosine function is at its lowest value (-1) and the amplitude is negative, so the minimums at the general representation of the function are given by:
y = 4*-1*cos(3(x + π/4)) + 2,
which simplifies to:
y = -4*cos(3(x + π/4)) + 2.
The maximums occur when the cosine function is at its highest value (1) and the amplitude is positive, so the maximums at the general representation of the function are given by:
y = 4*1*cos(3(x + π/4)) + 2,
which simplifies to:
y = 4*cos(3(x + π/4)) + 2.
To graph two full cycles, plot the minimums and maximums calculated above, and connect them with smooth curves. Label the x and y axes, as well as the minimums and maximums on the graph.