Graph f (x) = 3cos (4pix -pi/2) -2
f (x) = 3cos (4pix -pi/2) -2
= 3cos (4π(x - 1/8) ) - 2 <---- a horizontal shift of 1/8 to right
period = 2π/(4π) = 1/2
On the x-axis mark off 0, 1/8, 2/8, 3/8 and 4/8
the magnitude of your cosine curve is 3, so on the y-axis, mark off 5 or 6 units above and below
Knowing the general shape of a cosine curve, plot
(0,3), (1/8, 0) , (2/8,-1), (3/8,0) and (4/8, 3)
now move that curve 1/8 units to the right, then 2 units down
http://www.wolframalpha.com/input/?i=plot+y+%3D+3cos+(4%CF%80x)+,+y+%3D+3cos+(4pix+-pi%2F2)+-2,+y+%3D+3cos+(4%CF%80(x+-+1%2F8)+)+-+2
notice the first graph establishes the shape.
the second equation is as stated
the third equation in my simplied version
note that the 2nd and third graphs coincide.
To graph the function f(x) = 3cos(4πx - π/2) - 2, we can follow these steps:
Step 1: Determine the period
The period of a cosine function is given by the formula 2π/b, where b is the coefficient of x inside the cosine function. In this case, b = 4π, so the period is 2π/(4π) = 1/2.
Step 2: Determine the amplitude
The amplitude of a cosine function is given by the coefficient in front of the cosine function. In this case, the amplitude is 3.
Step 3: Determine the phase shift
The phase shift of a cosine function is given by the equation (c/b), where c is the constant inside the cosine function. In this case, c = -π/2 and b = 4π. So the phase shift is (-π/2)/(4π) = -1/8.
Step 4: Determine key points
To graph the function, we need to determine some key points. We can start with the x-intercepts, which occur when the cosine function equals zero:
3cos(4πx - π/2) - 2 = 0
3cos(4πx - π/2) = 2
cos(4πx - π/2) = 2/3
Using inverse cosine:
4πx - π/2 = cos^(-1)(2/3)
4πx = π/2 + cos^(-1)(2/3)
x = (π/2 + cos^(-1)(2/3))/(4π)
So we find the x-coordinate for the x-intercept.
Step 5: Plot the graph
Using the information we gathered from the previous steps, we can now plot the graph of f(x) = 3cos(4πx - π/2) - 2. Start by marking the x-intercepts, the maximum and minimum points (which occur at the amplitude), and any other points of interest (such as the y-intercept).
Remember that the period is 1/2, the amplitude is 3, and the phase shift is -1/8. Use these values to determine the shape of the graph and to plot additional points.
Once you have the key points, connect them smoothly to create the graph of f(x) = 3cos(4πx - π/2) - 2.