Question

Each function graphed is of the form y = c + cos x, y = c + sin x, y = cos(x - d), or y = sin(x - d), where d is the least possible positive value. Determine an equation of the graph.

Accepted Answer

Identify the key features of the graph such as amplitude, vertical shift, and phase shift. Since the functions are of the form $$y = c + \cos x$$, $$y = c + \sin x$$, $$y = \cos(x - d)$$, or $$y = \sin(x - d)$$, start by determining the vertical shift $$c by $$finding the midline of the graph (the average of the maximum and minimum values). Determine the amplitude of the graph by calculating half the distance between the maximum and minimum values. This amplitude corresponds to the coefficient in front of the sine or cosine function, which in this problem is 1, so confirm if the graph matches this amplitude. Next, decide whether the graph resembles a sine or cosine function by looking at the starting point (at $$x=0). If $$the graph starts at a maximum or minimum, it is likely a cosine function; if it starts at the midline going upward or downward, it is likely a sine function. Find the phase shift $$d by $$identifying the horizontal shift of the graph compared to the standard sine or cosine graph. The phase shift is the smallest positive value such that the function matches the graph, so measure how far the graph is shifted horizontally from the usual $$x=0$$ position. Write the equation by combining the vertical shift $$c$$, the function type (sine or cosine), and the phase shift $$d in $$the form $$y = c + \cos(x - d) or$$ $$y = c + \sin(x - d)$$, using the values you have determined.

Each function graphed is of the form y = c + cos x, y = c + sin x, y = cos(x - d), or y = sin(x - d), where d is the least possible positive value. Determine an equation of the graph.

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Key Concepts

Amplitude and Vertical Shift in Trigonometric Functions

Phase Shift in Sine and Cosine Functions

Identifying Function Type from Graph Shape

Each function graphed is of the form y = c + cos x, y = c + sin x, y = cos(x - d), or y = sin(x - d), where d is the least possible positive value. Determine an equation of the graph.<IMAGE>