Math Problem Statement

Determine the correct equation of the following graph.

Solution

The graph shows a periodic function, and it appears to be a sine or cosine function. To determine the correct equation, we need to analyze the graph's characteristics: amplitude, period, phase shift, and vertical shift.

  1. Amplitude: The maximum value is approximately 4, and the minimum value is approximately -4. Therefore, the amplitude is 4.

  2. Period: The distance between two consecutive peaks (or troughs) determines the period. From the graph, it appears that the period is 60 units (from -30 to 30 or from 0 to 60).

  3. Phase Shift: The graph appears to cross the y-axis at the origin (0,0), which suggests there is no horizontal shift for the cosine function.

  4. Vertical Shift: The graph oscillates around the x-axis (y=0), indicating there is no vertical shift.

Given these characteristics, the equation could be of the form y=Acos(Bx+C)+Dy = A \cos(Bx + C) + D.

Plugging in the values:

  • A=4A = 4 (amplitude)
  • B=2πperiod=2π60=π30B = \frac{2\pi}{\text{period}} = \frac{2\pi}{60} = \frac{\pi}{30} (since the period is 60)
  • C=0C = 0 (no phase shift)
  • D=0D = 0 (no vertical shift)

Thus, the equation of the graph is:

y=4cos(π30x)y = 4 \cos\left(\frac{\pi}{30} x\right)

Would you like further details or have any questions?

Here are five more questions you might want to ask:

  1. How do you determine the amplitude of a graph?
  2. What is the significance of the period in trigonometric functions?
  3. How do you find the phase shift of a function from its graph?
  4. What are the steps to convert a trigonometric function into its general form?
  5. Can you explain how vertical shifts affect the graph of a trigonometric function?

Tip: When identifying the equation of a trigonometric function from its graph, carefully note the amplitude, period, phase shift, and vertical shift, as these parameters define the function's behavior.

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Math Problem Analysis

Mathematical Concepts

Trigonometry

Formulas

Trigonometric function: y = A cos(Bx + C) + D

Theorems

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Suitable Grade Level

High School