Determine the simplest form of an equation for each graph. Choose b > 0, and include no phase shifts. (Midpoints and quarter points are identified by dots.)


<IMAGE>

Verified step by step guidance

1. Identify the type of trigonometric function represented by the graph (sine or cosine) by observing the starting point of the graph at x = 0. For example, if the graph starts at the midline going upward, it is likely a sine function; if it starts at a maximum or minimum, it is likely a cosine function.
2. Determine the amplitude (a) by measuring the vertical distance from the midline (horizontal center line) to the maximum or minimum point of the graph.
3. Find the period (T) by measuring the horizontal length of one complete cycle of the graph. Use the formula for the period of sine or cosine: \(T = \frac{2\pi}{b}\), and solve for \(b\) as \(b = \frac{2\pi}{T}\).
4. Since the problem specifies no phase shifts, set the phase shift \(c = 0\). Write the general form of the function as \(y = a \sin(bx)\) or \(y = a \cos(bx)\) depending on the function type identified in step 1.
5. Verify the function by checking key points such as midpoints and quarter points (where the function reaches zero, maximum, or minimum) to ensure the equation matches the graph's behavior without any horizontal shifts.

Key Concepts

Trigonometric Function Forms

Understanding the standard forms of sine and cosine functions, such as y = a sin(bx) or y = a cos(bx), is essential. These forms include amplitude (a), frequency (b), and phase shift, which define the shape and position of the graph. Recognizing these helps in writing the equation from the graph.

Amplitude and Period

Amplitude is the height from the midline to the peak of the wave, indicating the value of 'a'. The period is the length of one complete cycle, calculated as 2π divided by the frequency 'b'. Identifying these from the graph allows determination of 'a' and 'b' in the equation.

Phase Shift and Midline

Phase shift refers to horizontal shifts of the graph, but the question specifies no phase shifts, simplifying the equation. The midline is the horizontal axis around which the function oscillates, usually y=0 unless shifted vertically. Recognizing these helps in writing the simplest form without additional shifts.