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.)
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Identify the type of trigonometric function that best fits the graph, such as sine or cosine.
Determine the amplitude of the function by measuring the vertical distance from the midline to a peak or trough.
Find the period of the function by measuring the horizontal distance between two consecutive peaks or troughs.
Calculate the value of b using the formula for the period: .
Write the equation in the form or , using the amplitude and b value, ensuring there are no phase shifts.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Trigonometric Functions
Trigonometric functions, such as sine, cosine, and tangent, are fundamental in describing periodic phenomena. They relate angles to ratios of sides in right triangles and can be represented graphically. Understanding their properties, including amplitude, period, and frequency, is essential for analyzing and constructing equations that represent their graphs.
Amplitude refers to the maximum height of a wave from its midline, while the period is the distance over which the wave repeats itself. For sine and cosine functions, the amplitude is determined by the coefficient in front of the function, and the period is calculated using the formula 2π divided by the coefficient of the variable inside the function. These concepts are crucial for accurately modeling the behavior of trigonometric graphs.
Graphing trigonometric functions involves plotting points based on the function's values at specific angles. Key points include the midpoints and quarter points, which help in determining the shape and position of the graph. Understanding how to translate the properties of the function into a visual representation is vital for creating accurate equations that match the given graphs.