A cos wave has its first maximum at the point (pie/2, 2) and its first minimum at (5pie/2, -4) what is the equation of the function?
I remember writing 3cos(1/2(x-pie/2)-1 but I'm sure that it's wrong ?
To find the equation of the cosine function, we need to determine the amplitude, period, phase shift, and vertical shift.
First, let's analyze the given information. The first maximum occurs at (π/2, 2), which means the cosine function is at its highest point when x = π/2. The first minimum occurs at (5π/2, -4), which means the cosine function is at its lowest point when x = 5π/2.
The cosine function can be written as:
f(x) = A*cos(Bx + C) + D
where A represents the amplitude, B represents the frequency (1/period), C represents the phase shift, and D represents the vertical shift.
1. Amplitude (A): The amplitude is the distance from the center line to the highest or lowest point of the cosine graph. In this case, the highest point is 2 and the lowest point is -4, so the amplitude is (2 - (-4))/2 = 3.
Therefore, A = 3.
2. Period: The period of a cosine function is the distance between each cycle of the function. It can be calculated using the formula:
Period (T) = 2π/B
Since we know the highest point occurs at x = π/2 and the lowest point occurs at x = 5π/2, the distance between these two points is 4π. Therefore, the period is 4π.
Therefore, B = 2π/(4π) = 1/2.
3. Phase Shift (C): The phase shift is the horizontal shift of the graph. From the given information, we can see that the highest point occurs at x = π/2. The standard position for the first maximum of a cosine function is at x = 0. Therefore, there is a horizontal shift of π/2.
Therefore, C = -π/2.
4. Vertical Shift (D): The vertical shift is the amount by which the graph is moved up or down. From the given information, the center line is halfway between the highest and lowest points, which is (2 + (-4))/2 = -1.
Therefore, D = -1.
Now, we can substitute the values of A, B, C, and D into the equation:
f(x) = 3*cos((1/2)*(x - (-π/2))) - 1
Simplifying further, we get:
f(x) = 3*cos((1/2)*(x + π/2)) - 1
So the correct equation for the given cosine function is f(x) = 3*cos((1/2)*(x + π/2)) - 1.