A waveform is graphed on a coordinate plane. The x-axis ranges from negative 6 pi to 6 pi in increments of Start Fraction 3 pi over 2 End Fraction. The y-axis ranges from negative 3 to 3 in increments of 1.

Question

Given the graph, write the cosine function and explain how you arrived at your answer in 1–2 sentences.

Answer 1

To derive the cosine function from the given waveform, we first identify its amplitude, period, and phase shift from the graph. Assuming the waveform reaches a maximum of 3 and a minimum of -3, the amplitude is 3. If the period corresponds to the changes observed over the x-axis from -6π to 6π (which is 12π), the period can be calculated, with the cosine function having a standard form of \( y = a \cos(b(x - c)) + d \). Thus, the function can be expressed as \( y = 3 \cos\left(\frac{1}{2}x\right) \), where \(\frac{1}{2}x\) accounts for the period of 12π (since \(b\) is related to the period by \(2\pi/b = 12\pi\)).

The specifics, such as phase shift or whether it needs to be reflected, would depend on the exact shape of the waveform in the provided graph.