Trig Functions

Amplitude, Period, and Phase Shift

Given y = A sin(Bx + C) + D, identify amplitude, period, phase shift, and midline; build a key-point table for one complete period.

Key-point method: evaluate at x = phase_shift + i·(period/4) for i = 0, 1, 2, 3, 4 to find midline/max/midline/min/complete.

amplitude-period-phase For y = A sin(Bx + C) + D: Amplitude = |A| (vertical stretch from midline to peak). Period = 2π/|B| (horizontal length of one full cycle). Phase shift = -C/B (horizontal translation; positive = right). Midline: y = D (vertical shift of center line).

Example

Extract amplitude, period, phase shift, midline, and key points from a sine model.

highlighted = computed this step

Step 1 — Set up

Set up the expression.

y=2sin⁡(2x−π2)+1y= 2 \sin( 2 x- \frac{\pi}{2} )+ 1

Step 2 — Amplitude

Take the absolute value of A: |2| = 2.

∣A∣=∣2∣=2|A|=| 2 |= \hl{2}

Step 3 — Period

Use two pi divided by 2 for the period.

2π∣B∣=2π2=π\frac{ 2 \pi}{|B|}=\frac{ 2 \pi}{ 2 }= \hlmath{\pi}

Step 4 — Phase shift

Compute the phase shift: right pi over 4.

−CB=π4right-\frac{C}{B}= \hlmath{\frac{\pi}{4}} \quad \text{right}

Step 5 — Midline

Read the vertical shift: the midline is y = 1.

y=1y= \hl{1}

Step 6 — Key points

List the key points for one complete period.

xyroleπ41midline, risingπ23maximum3π41midline, fallingπ−1minimum5π41complete\hlmath{\begin{array}{c|c|c}x&y&\text{role}\\\frac{\pi}{4}&1&\text{midline, rising}\\\frac{\pi}{2}&3&\text{maximum}\\\frac{3\pi}{4}&1&\text{midline, falling}\\\pi&-1&\text{minimum}\\\frac{5\pi}{4}&1&\text{complete}\end{array}}