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 step by step guidance

1. Step 1: Identify the type of trigonometric function. Look at the graph to determine if it resembles a sine or cosine function.
2. Step 2: Determine the vertical shift 'c'. Check if the midline of the graph is shifted up or down from the x-axis.
3. Step 3: Identify any horizontal shift 'd'. Observe if the graph is shifted left or right from the standard position of the sine or cosine function.
4. Step 4: Write the equation using the identified values of 'c' and 'd'. For example, if it's a cosine function with a vertical shift, the equation could be y = c + cos(x - d).
5. Step 5: Verify the equation by checking if it matches the key points and features of the graph, such as amplitude, period, and phase shift.

Key Concepts

Trigonometric Functions

Trigonometric functions, such as sine and cosine, describe periodic phenomena and are defined based on the unit circle. The functions y = sin(x) and y = cos(x) oscillate between -1 and 1, with specific properties such as amplitude, period, and phase shift. Understanding these functions is essential for analyzing their transformations and behaviors in graphs.

Vertical and Horizontal Shifts

Vertical and horizontal shifts refer to the transformations applied to the basic sine and cosine functions. A vertical shift occurs when a constant 'c' is added to the function, moving the graph up or down. A horizontal shift, represented by 'd' in the equations, adjusts the graph left or right, affecting the starting point of the wave without altering its shape.

Phase Shift

Phase shift is the horizontal displacement of a periodic function, which is determined by the value 'd' in the equations y = cos(x - d) or y = sin(x - d). This shift indicates how much the graph is moved along the x-axis, impacting the alignment of the peaks and troughs of the wave. Understanding phase shifts is crucial for accurately determining the equation of a transformed trigonometric graph.