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 average of the maximum and minimum values). Determine 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 period of the function by measuring the distance between two consecutive peaks or troughs. The standard period for sine and cosine is $$2\pi$$, so use this to confirm if there is any horizontal stretching or compression. Calculate the phase shift $$d by $$identifying the horizontal shift from the standard sine or cosine graph. The phase shift is the smallest positive value such that the function matches the graph, and it appears inside the function argument as $$x - d. $$Write the equation by combining the vertical shift $$c$$, the function type (sine or cosine), and the phase shift $$d$$ into 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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Concetti chiave

Amplitude and Vertical Shift in Trigonometric Functions

Phase Shift and Horizontal Translation

Identifying the Base Function (Sine vs Cosine)

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>