Given: y = 2 cos 1/2 (theta + pi/4) + 1
What is the number of cycles?
what is the horizontal shift?
number of cycles per what ?
your curve is a horizontal shift of π/4 to the left of the curve
y = 2 cos (Ø/2) + 1
I'm not sure about the number of cycles, I think I'll assume it means the number of curves since there are 4 of them if i graph. Here is what I've already answered:
a. amplitude = 2
b. number of cycles = 4?
c. (is the phase shift also horizontal shift?) Phase shift = -(pi/4)
d. vertical shift = 1
For the normal domain of 9 ≤ Ø ≤ 2π
period = 2π/(1/2) = 4π
so number of periods in the domain = 2π/4π = 1/2
look at Wofram's graph to confirm this
http://www.wolframalpha.com/input/?i=plot+y+%3D+2+cos+(1%2F2+(x+%2B+%CF%80%2F4))+%2B+1
Notice from a max to a min, which would be 1/2 of a period shows up as appr 6 (2π = appr 6.28)
To find the number of cycles and the horizontal shift of the given function, we need to analyze its equation, y = 2cos(1/2(theta + pi/4)) + 1.
First, let's rewrite the equation in the general form of a cosine function: y = A*cos(B(theta - C)) + D.
Comparing the given equation with the general form, we can see that:
A = 2, which represents the amplitude (the maximum value of the function).
B = 1/2, which represents the frequency (how many cycles are squeezed into a 2π interval).
C = -pi/4, which represents the horizontal shift (the amount by which the function is moved left or right).
D = 1, which represents the vertical shift (the amount by which the function is moved up or down).
Number of cycles:
To find the number of cycles, we need to divide the coefficient of theta (B) by the absolute value of 2π.
Number of cycles = |B| / (2π) = (1/2) / (2π) = 1 / (4π)
Therefore, the number of cycles for the given function is 1 / (4π).
Horizontal shift:
To find the horizontal shift, we need to determine the value of C, which represents the amount of shift.
C = -pi/4
Therefore, the horizontal shift for the given function is -pi/4.