Math Problem Statement

solve

Write an equation for the solid blue curve in the graph.

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 x=7π6,7π2,and14π3x = \frac{7\pi}{6}, \frac{7\pi}{2}, \text{and} \frac{14\pi}{3}, 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:

  1. 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.

  2. 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: y=Acsc(BxC)+Dy = A \csc(Bx - C) + D We will now determine the constants AA, BB, CC, and DD.

  3. 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, A=7A = 7 and D=0D = 0.

  4. Period (B): The period of the sine and cosecant functions is given by 2πB\frac{2\pi}{B}. From the graph, the period seems to be the distance between consecutive vertical asymptotes, which is 14π37π6=7π2\frac{14\pi}{3} - \frac{7\pi}{6} = \frac{7\pi}{2}. Solving for BB: 2πB=7π2    B=47\frac{2\pi}{B} = \frac{7\pi}{2} \implies B = \frac{4}{7}

  5. Phase Shift (C): The graph does not seem to be shifted horizontally, so C=0C = 0.

Final Equation:

Thus, the equation of the solid blue curve is: y=7csc(47x)y = 7 \csc\left(\frac{4}{7}x\right)

Would you like further clarification on any steps?

Related Questions:

  1. How would the equation change if the vertical shift were non-zero?
  2. What is the effect of changing the amplitude AA in the equation?
  3. How does the period of a cosecant function relate to that of a sine function?
  4. How can you derive an equation for the dashed red curve?
  5. 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