what would be the period of a graph with an amplitude of 1 midline of one and horizontel of pi/4 and pi/2 etc? Explain the period.
The period of a graph is the distance between one complete cycle of the graph. In other words, it is the length of the interval over which the graph repeats itself.
For a graph with a horizontal shift of pi/4, pi/2, etc. (also known as the phase shift), the period can be determined by the formula:
Period = 2π/b
where b is the coefficient of x in the equation of the graph. In this case, since the graph is a sine or cosine function, the equation will have the form:
y = a*sin(bx) + c
where a represents the amplitude, b represents 2π/period, and c is the midline (or vertical shift).
Given that the amplitude (a) is 1 and the midline (c) is 1, the equation becomes:
y = sin(bx) + 1
To find the period, we need to determine the value of b. The coefficient of x in this case is 1, so b = 2π/period. Rearranging the equation, we have:
sin(bx) + 1 = y
sin(x/period) + 1 = y
Since the midline is at 1, the maximum value of y would be 2 (1 + amplitude). Therefore, the minimum value for sin(x/period) would be 1 (when y = 2), which occurs when x = (2n - 1)π/2, where n represents an integer.
Hence, the graph would complete one cycle from x = (2n - 1)π/2 to x = (2n + 1)π/2. The distance between these two points represents the period of the graph.
Therefore, the period of the graph with an amplitude of 1, midline of 1, and horizontal shift of pi/4, pi/2, etc. is:
Period = [(2n + 1)π/2] - [(2n - 1)π/2]
= π/2
Thus, the period is π/2.