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 $$y = \sin(x - d)$$, determine the vertical shift $$c by $$finding the midline of the graph (the horizontal line halfway between the maximum and minimum values). Determine whether the graph resembles a sine or cosine function by looking at the starting point at $$x=0. $$For $$y = \cos x$$, the graph starts at a maximum value when $$x=0$$, while for $$y = \sin x$$, the graph starts at zero and increases or decreases depending on the phase shift. Find the phase shift $$d by $$identifying the horizontal shift of the graph from the standard sine or cosine graph. The phase shift $$d is $$the smallest positive value such that the function matches the graph's horizontal displacement. Write the equation using the identified vertical shift $$c$$ and phase shift $$d. If $$the graph matches a cosine function, use $$y = c + \cos(x - d)$$; if it matches a sine function, use $$y = c + \sin(x - d). If $$there is no horizontal shift, $$d$$ will be zero. Verify your equation by checking if it correctly predicts key points on the graph such as maxima, minima, and zeros, ensuring the amplitude and vertical shift match the graph's features.

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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Verified video answer for a similar problem:

Key Concepts

Amplitude and Vertical Shift in Trigonometric Functions

Phase Shift in Sine and Cosine Functions

Identifying the Function Type from Graph Characteristics

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>