Math Problem Statement

Retrieve all coefficients by the graph of this pictured general sine function  f(x) = r sin(ωx + θ_0) + c

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 f(x)=rsin(ωx+θ0)+cf(x) = r \sin(\omega x + \theta_0) + c from the given graph, let's analyze the key features of the sine wave in the image.

Observations from the Graph:

  1. Amplitude rr: This is the maximum displacement from the centerline of the wave. Observing the graph, the wave oscillates between approximately y=1y = 1 and y=7y = 7, so the amplitude rr can be calculated as: r=712=3r = \frac{7 - 1}{2} = 3

  2. Vertical Shift cc: 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: c=7+12=4c = \frac{7 + 1}{2} = 4

  3. Angular Frequency ω\omega: To determine ω\omega, we need the period TT of the sine wave, which is the distance between two successive peaks. Observing the x-axis, the period is approximately T=8T = 8 units. Since the period TT is related to ω\omega by the formula T=2πωT = \frac{2\pi}{\omega}, we can solve for ω\omega: ω=2πT=2π8=π4\omega = \frac{2\pi}{T} = \frac{2\pi}{8} = \frac{\pi}{4}

  4. Phase Shift θ0\theta_0: The phase shift θ0\theta_0 determines the horizontal translation of the function. Observing the graph, the wave crosses the centerline at a point between x=3x = 3 and x=4x = 4. However, the precise value of θ0\theta_0 may require more details. For simplicity, if we assume the wave starts at x=0x = 0 without a phase shift, we could take θ0=0\theta_0 = 0.

Final Coefficients

Thus, based on the visual inspection, the estimated coefficients for the function f(x)=rsin(ωx+θ0)+cf(x) = r \sin(\omega x + \theta_0) + c are:

  • Amplitude rr: 3
  • Vertical Shift cc: 4
  • Angular Frequency ω\omega: π4\frac{\pi}{4}
  • Phase Shift θ0\theta_0: 0 (assuming no shift; adjust if additional information suggests otherwise)

Function Form

The function based on these coefficients would be: f(x)=3sin(π4x)+4f(x) = 3 \sin\left(\frac{\pi}{4} x\right) + 4

Would you like further clarification on any part of this solution?

Additional Questions:

  1. How does changing the amplitude affect the graph of a sine function?
  2. What effect does the vertical shift cc have on the sine wave's position?
  3. How is the period of a sine function related to its angular frequency?
  4. Why might the phase shift θ0\theta_0 be significant in certain applications?
  5. 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