Transform and reconstruct sine and cosine functions

Read the factored argument

For y=R sin(n(x−h))+k with R>0 and n>0, amplitude is R, period is 2π/n and the range for all real x is [k−R,k+R]. The horizontal shift is h, not nh. The cosine form has the same amplitude, period and range

For y=5 sin(2x+π/3)−1, find the amplitude

For y=5 sin(2x+π/3)−1, give the coefficient of π in the period

Write 2x+π/3=2(x−h). Give h/π

Find the global minimum of y=5 sin(2x+π/3)−1.

Can sin2x+cos3x generally be combined as R sin(2x+φ) with constant R and φ?

Sketch a full period, ungraded

Sketch one period of y=5 sin(2(x+π/6))−1. Label the mean line, range and five quarter-period points. Explain the horizontal shift and why a restricted interval might not attain the global extrema

Model: use the phase anchor

For y=5 sin(2(x+π/6))−1, starting at x=−π/6, the five points are (−π/6,−1), (π/12,4), (π/3,−1), (7π/12,−6), (5π/6,−1); consecutive x-values differ by π/4; the mean line is y=−1 and the full-cycle range is [−6,4]; a shorter restricted domain can omit the maximum or minimum