Questions: Find a function of the form y=A sin (k x)+C or y=A cos (k x)+C whose graph matches the function shown below: Leave your answer in exact form; if necessary, type pi for pi.

Solution Steps

The amplitude \( A \) is the maximum value of the function from the midline. From the graph, the maximum value is 4 and the minimum value is -4. Therefore, the amplitude \( A \) is: \[ A = 4 \]

The period \( T \) is the distance over which the function repeats. From the graph, the function completes one full cycle from \( x = -12 \) to \( x = 0 \), so the period \( T \) is: \[ T = 12 \]

The frequency \( k \) is related to the period by the formula: \[ k = \frac{2\pi}{T} \] Substituting the period \( T = 12 \): \[ k = \frac{2\pi}{12} = \frac{\pi}{6} \]

The vertical shift \( C \) is the midline of the function. From the graph, the midline is at \( y = 0 \), so: \[ C = 0 \]

The graph starts at the midline and goes downwards, which is characteristic of a cosine function with a phase shift. However, since the phase shift is not given, we can use the sine function which starts at the midline and goes upwards or downwards. Here, it goes downwards, so we use a negative sine function: \[ y = -A \sin(kx) + C \]

Final Answer