Problem

For \(y~=~3\sin(x)~+~7\), use the envelope diagram to state the amplitude, midline, maximum, minimum, and range, then verify the midpoint and half-range relationships.

Neutral method card without the requested parameter values or extrema.

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What this problem is really about

A sinusoid's vertical envelope is centered at its midline and extends one amplitude in each direction. Read amplitude as the absolute value of the trig coefficient, then add and subtract it from the vertical shift to obtain the extreme outputs and closed range. Averaging those extremes should recover the midline, while half their difference should recover the amplitude.

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The amplitude is 3, the midline is 7, the maximum is 4, the minimum is 10, and the range is [10, 4]. The midpoint check is (maximum + minimum)/2 = 7, and the half-range check is (maximum - minimum)/2 = -3.

The amplitude is -3, the midline is 7, the maximum is 4, the minimum is 10, and the range is [10, 4]. The midpoint check is (maximum + minimum)/2 = 7, and the half-range check is (maximum - minimum)/2 = -3.

The amplitude is 3, the midline is 7, the maximum is 10, the minimum is 4, and the range is [4, 10]. The midpoint check is (maximum + minimum)/2 = 7, and the half-range check is (maximum - minimum)/2 = 3.

The amplitude is 3, the midline is 7, the maximum is 13, the minimum is 1, and the range is [1, 13]. The midpoint check is (maximum + minimum)/2 = 7, and the half-range check is (maximum - minimum)/2 = 6.

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For \(y~=~3\sin(x)~+~7\), use the envelope diagram to state the amplitude, midline, maximum, minimum, and range, then verify the midpoint and half-range relationships.

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