LEMMA

Advanced trigonometry

Amplitude, period, and phase

ALEKS placement

Amplitude controls height, the horizontal factor controls period, and the phase shift appears after you factor that number out.

What this covers

y=Asin(B(x−C))+D,period=∣B∣2π​
  • Amplitude is |A|, and a negative A flips the wave.
  • Period is 2π/|B| for sine and cosine, π/|B| for tangent.
  • Factor B out of the parentheses before reading the phase shift C.
  • D is the midline.

Worked example

Worked example

Find the amplitude, period, and shift of y = 3sin(2x - π).

  1. y=3sin(2(x−2π​))Factor the 2 out before reading the shift.
  2. A=3,period=π,shift=2π​rightRead each parameter off the factored form.
Another worked example

Find the amplitude, period, phase shift, and midline of y = -2cos((x + π)/2) + 1.

  1. y=−2cos(21​(x−(−π)))+1Write the inside as B(x - C) before reading the shift.
  2. amplitude=2,period=1/22π​=4πUse the absolute vertical scale for amplitude and divide by the horizontal factor for period.
  3. shift=−π,midline:y=1C is negative π, so the graph shifts left π. D lifts its midline to 1.

A common mistake

A mistake Lemma catches

From

2x−π=2(x−π)2x−π=2(x−2π​)

“Factor the 2 out of the whole inside: 2x − π = 2(x − π/2), since 2 · π/2 = π. The shift is π/2 to the right. Reading π straight off 2x − π skips the factoring: 2(x − π) = 2x − 2π, which is not the inside.”

Try one

Sample problem

Find the exact period.

y=cos(6x)

Practice trig graphs free

No account, no email. Every line you type is checked by the same computer algebra system as the worked example above.

Where this fits

Before this