Which equation best describes this function?

Question

BlackTom AI · Accepted Answer

We start by identifying the key features of the graph: a midline around y = 2, an amplitude about 1.5 (the peaks reach roughly 3.5 and the troughs around 0.5), and a periodic wave with several full cycles between 0 and 2π. This suggests a sinusoidal function of the form y = 2 ± 1.5·something, where the ± indicates whether the chosen trigonometric function is cosine or sine and how its phase aligns with the graph.

Option a: y = -1.5 cos(3(x + π/6)) + 2. Here the amplitude is 1.5, the vertical shift is 2, and the sign is negative, which would invert the cosine. The phase shift of +π/6 inside the argument could align peaks, but the overall negative amplitude would place maxima where cos(·) is -1, potentially shifting the peaks by half a period compared to a positive amplitude. We need to compare with the graph’s peak positions; if the graph shows a peak where cos(3x + phase) would be +1, a negative amplitude would flip that to a trough, which may not match.

Option b: y = -1.5 cos(3x + π/6) + 2. Similar to option a but without the outer shift, this also uses a negative amplitude. The lack of an explicit (x + ...) inside the product changes where the phase lands relative to x. Depending on the actual peak placements in the graph, this might not align as well as a positive-amplitude form.

Option c: y = 1.5 cos(3(x + π/6)) + 2. This uses a positive amplitude with a phase shift of π/6 inside the argument, and a vertical shift of 2. A positive amplitude places maxima where the cosine argument is an integer multiple of 2π. The phase shift helps align the peaks with the observed positions on the x-axis, and the midline at 2 with an amplitude of 1.5 matches the visual height of the oscillation seen in the plot. Overall, this option seems to fit the described midline, amplitude, and peak alignment.

Option d: y = -1.5 sin(3(x + π/6)) + 2. This form uses a sine with a negative amplitude and a phase shift. Since sine is a cosine-shifted function, it could also describe the same wave, but the specific combination here would produce a particular phase relationship that might or might not match the plotted peaks and troughs as cleanly as the cosine-based representation in option c. Without explicit comparison of peak positions, this is less direct than option c for matching the given graph.

In summary, the graph displays a midline at y = 2, amplitude about 1.5, and peaks/valleys arranged in a way that a positive-amplitude cosine with an appropriate phase shift best aligns with the observed x-values of maxima and minima. Among the choices, option c provides a positive amplitude with a phase shift that can place the peaks at the correct x-coordinates while maintaining the correct vertical shift and amplitude. Therefore, option c is the most consistent with the graph’s features.

MHF4U - Advanced Functions 12 (2025-26) - A

Which equation best describes this function?

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