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 values of a and b found in the previous steps.
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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 are essential for modeling wave-like behaviors in graphs. Understanding their properties, including amplitude, period, and frequency, is crucial for determining the simplest form of an equation that represents a given graph.
Amplitude refers to the height of the wave from its midline to its peak, while the period is the distance over which the wave repeats itself. In trigonometric equations, the amplitude is determined by the coefficient in front of the sine or cosine function, and the period is calculated using the formula 2π divided by the frequency. Recognizing these characteristics helps in accurately formulating the equation that matches the graph.
Graphing trigonometric functions involves plotting points based on the function's values at specific angles. Key points, such as midpoints and quarter points, help in sketching the graph accurately. Understanding how to identify these points and their corresponding values is essential for translating the visual representation of the graph into a mathematical equation.