Questions: Question 14 The graph below shows one complete cycle of the graph of an equation containing a trigonometric function. Find an equation to match the graph.

Solution Steps

The graph shows a sinusoidal wave, which suggests it could be a sine or cosine function. The wave starts at \( y = 0 \), reaches a maximum at \( y = 1 \), a minimum at \( y = -1 \), and returns to \( y = 0 \). This pattern is characteristic of a sine function.

The amplitude of the wave is the maximum value of \( y \), which is 1. Therefore, the amplitude \( A \) is 1.

The period of the function is the length of one complete cycle. From the graph, one complete cycle occurs from \( x = 0 \) to \( x = 6\pi \). Therefore, the period \( T \) is \( 6\pi \).

The frequency \( B \) is related to the period by the formula \( T = \frac{2\pi}{B} \). Solving for \( B \): \[ 6\pi = \frac{2\pi}{B} \] \[ B = \frac{2\pi}{6\pi} = \frac{1}{3} \]

The general form of the sine function is \( y = A \sin(Bx + C) + D \). Since there is no vertical shift (\( D = 0 \)) and no horizontal shift (\( C = 0 \)), the equation simplifies to: \[ y = \sin\left(\frac{1}{3}x\right) \]

Final Answer