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Sine transformations: amplitude, period and phase

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      Trigonometry · Transformations

      Question

      Using y = 2sin(2x + π/4), illustrate the amplitude, period and phase transformations of y = A sin(ωx + φ).

      Sine function transformationsCompare y = sin x with y = 2sin(2x + π/4), marking amplitude 2, period π and the horizontal shift.OxyA = 2Baseline y = sin xTarget y = 2sin(2x + π/4)Period T = 2π / ω = π
      The dashed curve is y = sin x; the solid curve is y = 2sin(2x + π/4), with amplitude 2 and period π.

      Step-by-step solution

      1. Identify the parameters

      Here A controls vertical scaling, ω controls the period T = 2π/ω, and φ is the phase constant. For this example A and ω are positive.

      2. Translate the baseline graph

      Starting from y = sin x, replace x with x + π/4. This shifts the graph left by π/4 and gives y = sin(x + π/4).

      3. Compress horizontally

      Replace x with 2x in the translated function. Horizontal coordinates are halved, giving y = sin(2x + π/4). The period becomes 2π/2 = π. The resulting horizontal shift from the original sine graph is −φ/ω = −π/8.

      4. Stretch vertically

      Multiply all function values by 2. The amplitude is 2, the range is [−2, 2], and the final graph is y = 2sin(2x + π/4).

      The parameters are A = 2; T = π; φ = π/4.

      Which method does this problem use?

      Keep track of transformation order. In this construction, translate first, then scale horizontally, then vertically. The phase constant φ is not itself the final horizontal displacement when ω differs from 1.