📝 Trigonometric Graphs: Cycles and Amplitude

TL;DR

Key points

• •A cycle covers one period and can be repeated to reconstruct the graph across its domain. The instructor uses basic cycles beginning at x = 0 and divides each into four equal intervals.
• •Sine, cosine, secant, and cosecant have period 2π; tangent and cotangent have period π.
• •Sine and cosine have amplitude 1, with maximum 1 and minimum −1. Their basic cycles use quarter points at π/2, π, 3π/2, and 2π.
• •Cosecant is the reciprocal of sine, with vertical asymptotes at multiples of π. Secant is the reciprocal of cosine, with vertical asymptotes at odd multiples of π/2.
• •Secant and cosecant have relative extrema at y = ±1, but no absolute maximum or minimum. The instructor calls these key locations 'one points' rather than using amplitude.
• •For a tangent cycle from 0 to π, the vertical asymptote is at π/2, and the key points are (π/4, 1) and (3π/4, −1). Cotangent has asymptotes at the endpoints and passes through the same two key points.
• •For y = a·f(bx + c), a changes key y-values from ±1 to ±a; a negative value reflects the graph across the x-axis.
• •The transformed period is the original period divided by b. The phase shift is −c/b, which moves the basic cycle's starting x-coordinate from 0 to −c/b.
• •To locate transformed quarter points, start at the phase shift and repeatedly add one quarter of the new period.
• •The example y = 3 sin x changes the amplitude to 3 without changing the period or starting point. For y = 3 sin(2x), the period becomes π and quarter-point spacing becomes π/4.
• •For y = 3 sin(2x − π), the amplitude is 3, the period is π, and the phase shift is π/2 to the right. The cycle runs from π/2 to 3π/2, with intermediate quarter points at 3π/4, π, and 5π/4.
• •Acoustics and noise filtering were used to illustrate why recognizing and manipulating wave amplitudes matters.

Decisions

• •The instructor restricted the graphs under consideration to y = a·f(bx + c), with b positive. A corrected handout including this restriction had been uploaded.
• •The instructor requires coordinates to be labeled for important points off the axes; points on the x- or y-axis do not need coordinate labels.
• •The instructor chose to work on exercise 5 next, then return to exercise 4 because it contains a problem to examine.

Action items

• •Attempt the remaining exercises without Desmos, then check the answers.
• •Work through exercise 5, then return to examine the problem in exercise 4.

Open questions

• •How can Desmos be forced to display the desired x-axis markings? The instructor had not found a method, and the older graphing program no longer works on Windows 11.
• •What is the problem in exercise 4? The instructor flagged it but did not explain it in this transcript.