For y=|asin bx+c| or y=|acos bx+c|, first plot the inner trigonometric graph on the stated domain, then reflect its negative heights. Inputs stay fixed. Find endpoints, zero crossings and maxima/minima using familiar exact angles. Use these points to decide which intervals are negative. Reflect each such interval, retaining all zeros. For y=|sin x|, 0 <= x 2pi, zeros are 0,pi,2pi and peaks are (pi2,1),(3pi2,1). The second half-wave reflects because sine is negative there. Check: a vertical shift inside the modulus can change the number of zeros. The range of |sin x+2| is [1,3], not [0,3]. No double-angle identity is needed to sketch |sin2x|.

For \(y=|a\sin bx+c|\) or \(y=|a\cos bx+c|\), first plot the inner trigonometric graph on the stated domain, then reflect its negative heights. Inputs stay fixed.

Find endpoints, zero crossings and maxima/minima using familiar exact angles. Use these points to decide which intervals are negative. Reflect each such interval, retaining all zeros.

For \(y=|\sin x|\), \(0\le x\le2\pi\), zeros are \(0,\pi,2\pi\) and peaks are \(\left(\dfrac{\pi}{2},1\right),\left(\dfrac{3\pi}{2},1\right)\). The second half-wave reflects because sine is negative there.

Check: a vertical shift inside the modulus can change the number of zeros. The range of \(|\sin x+2|\) is \([1,3]\), not \([0,3]\). No double-angle identity is needed to sketch \(|\sin2x|\).

Modulus graphs of sine and cosine

Modulus graphs of sine and cosine