Math Problem Statement
Retrieve all coefficients by the graph of this pictured general sine function f(x) = r sin(ωx + θ_0) + c
Solution
To retrieve the coefficients of the general sine function from the given graph, let's analyze the key features of the sine wave in the image.
Observations from the Graph:
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Amplitude : This is the maximum displacement from the centerline of the wave. Observing the graph, the wave oscillates between approximately and , so the amplitude can be calculated as:
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Vertical Shift : This is the midpoint of the maximum and minimum values of the sine wave, representing the vertical displacement of the wave. Given the maximum and minimum values:
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Angular Frequency : To determine , we need the period of the sine wave, which is the distance between two successive peaks. Observing the x-axis, the period is approximately units. Since the period is related to by the formula , we can solve for :
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Phase Shift : The phase shift determines the horizontal translation of the function. Observing the graph, the wave crosses the centerline at a point between and . However, the precise value of may require more details. For simplicity, if we assume the wave starts at without a phase shift, we could take .
Final Coefficients
Thus, based on the visual inspection, the estimated coefficients for the function are:
- Amplitude : 3
- Vertical Shift : 4
- Angular Frequency :
- Phase Shift : 0 (assuming no shift; adjust if additional information suggests otherwise)
Function Form
The function based on these coefficients would be:
Would you like further clarification on any part of this solution?
Additional Questions:
- How does changing the amplitude affect the graph of a sine function?
- What effect does the vertical shift have on the sine wave's position?
- How is the period of a sine function related to its angular frequency?
- Why might the phase shift be significant in certain applications?
- Can you derive a similar function if the period was doubled?
Tip:
When analyzing sine or cosine functions, identifying key points like the maximum, minimum, and centerline crossings can provide insights into the function's parameters.
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Math Problem Analysis
Mathematical Concepts
Trigonometry
Sine function
Graph analysis
Formulas
f(x) = r sin(ωx + θ_0) + c
Amplitude r = (y_max - y_min) / 2
Vertical shift c = (y_max + y_min) / 2
Angular frequency ω = 2π / T
Theorems
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Suitable Grade Level
Grades 10-12