Math Problem Statement

Determine the period of the sinusoidal function given its maximums and minimums at specific points.

Solution

To determine the period of the sinusoidal function, we analyze the given data:

Key points from the problem:

  1. The function has maximums at (5π7,1)\left(-\frac{5\pi}{7}, 1\right) and (3π7,1)\left(-\frac{3\pi}{7}, 1\right).
  2. The function has a minimum at (4π7,1)\left(-\frac{4\pi}{7}, -1\right).
  3. The xx-axis is in radians, and the function is sinusoidal.

Step-by-step solution:

  1. Distance between two maximums:
    The distance between consecutive maximum points on a sinusoidal graph is half the period of the function.

    The xx-coordinates of the maximum points are 5π7-\frac{5\pi}{7} and 3π7-\frac{3\pi}{7}.
    The distance between these points is: 3π7(5π7)=3π7+5π7=2π7.\left| -\frac{3\pi}{7} - \left(-\frac{5\pi}{7}\right) \right| = \left| -\frac{3\pi}{7} + \frac{5\pi}{7} \right| = \frac{2\pi}{7}.

    Hence, the distance between consecutive maximums is 2π7\frac{2\pi}{7}, which equals half the period.

  2. Calculate the period:
    Since the distance between consecutive maximums is T2\frac{T}{2}, where TT is the period, we find: T=22π7=4π7.T = 2 \cdot \frac{2\pi}{7} = \frac{4\pi}{7}.

Final Answer:

The period of the sinusoidal function is: 4π7\boxed{\frac{4\pi}{7}}

Would you like a more detailed explanation or visual representation of the solution?

Here are 5 related questions for practice:

  1. How would the period change if the function were horizontally stretched instead of compressed?
  2. What is the amplitude of the function given the maximum and minimum values?
  3. How would you determine the phase shift of this sinusoidal function?
  4. How do you write the equation of the function in terms of sine or cosine?
  5. What is the frequency of the function in terms of 1period\frac{1}{\text{period}}?

Tip: When dealing with sinusoidal functions, always consider how the period is related to the distance between peaks or troughs!

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Math Problem Analysis

Mathematical Concepts

Sinusoidal Functions
Trigonometry
Periodicity

Formulas

Period = 2 * (distance between consecutive maximums)

Theorems

Periodicity of Sinusoidal Functions

Suitable Grade Level

Grades 9-11