Problem

Compare \(f(x)~=~2\sin(x)~+~1\) and \(g(x)~=~3\sin(x)~-~2\) by amplitude, period, phase, midline, range, reflection, and starting pattern.

Two sine waves over 0 through 2 pi with different amplitudes and midlines.

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What this problem is really about

Organize the comparison by parameter rather than describing one entire wave at a time. The outside coefficient controls amplitude and reflection, the vertical shift sets the midline and range, and the inside expression controls period and phase. Matching those rows also reveals whether the two waves share a starting pattern.

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f has amplitude 2, period 2π, midline y = 1, and range [−1, 3]. g has amplitude 3, period π and a different phase, midline y = −2, and range [−5, 1]; both start as unreflected sine.

f has amplitude 2, period 2π, midline y = 1, and range [−1, 3]. g has amplitude 3, the same period and phase, midline y = −2, and range [−5, 1]; both start as unreflected sine.

f has amplitude 1, period 2π, midline y = 1, and range [−1, 3]. g has amplitude 3, the same period and phase, midline y = −2, and range [−5, 1]; both start as unreflected sine.

f has amplitude 2, period 2π, midline y = 1, and range [−1, 3]. g has amplitude −3, the same period and phase, midline y = −2, and range [−5, 1]; both start as unreflected sine.

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Compare \(f(x)~=~2\sin(x)~+~1\) and \(g(x)~=~3\sin(x)~-~2\) by amplitude, period, phase, midline, range, reflection, and starting pattern.

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