Math Problem Statement
solve
Solution
From the image, we observe a periodic function with vertical asymptotes and some symmetry. The solid blue curve resembles the graph of a tangent or cotangent function, with repeated intervals and vertical asymptotes. The vertical asymptotes are located at , suggesting that the function could be related to the cosecant function, as it typically has vertical asymptotes where sine equals zero.
Step-by-step Analysis:
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Periodicity and Asymptotes: The vertical asymptotes suggest a cosecant function. As cosecant is the reciprocal of sine, its vertical asymptotes occur where sine equals zero, which matches the points seen in the graph.
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Equation Format: Based on the periodic behavior and location of the asymptotes, the equation for the solid blue curve appears to be of the form: We will now determine the constants , , , and .
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Amplitude and Vertical Shift (A and D): The curve oscillates between 7 and -7, meaning that the amplitude is approximately 7 and there is no vertical shift. Hence, and .
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Period (B): The period of the sine and cosecant functions is given by . From the graph, the period seems to be the distance between consecutive vertical asymptotes, which is . Solving for :
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Phase Shift (C): The graph does not seem to be shifted horizontally, so .
Final Equation:
Thus, the equation of the solid blue curve is:
Would you like further clarification on any steps?
Related Questions:
- How would the equation change if the vertical shift were non-zero?
- What is the effect of changing the amplitude in the equation?
- How does the period of a cosecant function relate to that of a sine function?
- How can you derive an equation for the dashed red curve?
- How would you sketch a cotangent function with similar features?
Tip:
When analyzing trigonometric functions, always check for key points like asymptotes and intercepts to identify the type of function and its transformations.
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Math Problem Analysis
Mathematical Concepts
Trigonometry
Cosecant Function
Asymptotes
Periodicity
Formulas
y = A * csc(Bx - C) + D
Theorems
Cosecant function has vertical asymptotes where sine equals zero.
Suitable Grade Level
Grades 10-12
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