Twelfth Grade · Math · week 19

Graphing sine and cosine: amplitude, period, and phase shift

Graph transformed sine and cosine functions by identifying amplitude, period, phase shift, and vertical shift, and locate key points on one full cycle.

The graphs of y=sin(x) and y=cos(x) are smooth waves built directly from the unit circle: literally plot (angle, sin(angle)) as the angle sweeps around, and the wave appears. Both complete one full cycle every 2*pi and stay within [-1,1] in their basic form.

Four numbers describe any transformed wave y = a*sin(b(x-h)) + k (or cosine): amplitude |a| (how far the wave rises and falls from its center), period 2*pi/b (the length of one full cycle -- a bigger b means a faster, shorter cycle), phase shift h (how far the whole wave slides left or right), and vertical shift k (where the center line sits). A negative a flips the wave upside down.

📋 For the teacher
Teach it (10–15 min):
  • Build the wave together by literally plotting the unit-circle sine values at 0, pi/2, pi, 3pi/2, and 2pi and connecting them smoothly, so the shape feels earned.
  • Isolate amplitude and period as two separate questions living in two separate locations: amplitude multiplies the WHOLE function (outside); period comes from what multiplies x (inside).
  • Find where a cycle STARTS by setting the inside expression equal to 0 and solving for x -- that single x-value anchors every other key point.
  • Have them predict a graph's shape from its equation before plotting a single point, then check the prediction -- this builds the skill of reading equations, not just plotting them.
Talk about it:
  • Where in the equation do you find the amplitude, and where do you find the period?
  • Why does a coefficient of x INSIDE the function change how fast the wave repeats, while a coefficient OUTSIDE changes how tall it is?
  • If you know a wave's period, how would you find the coefficient of x that produces it?

Mastery looks like: They identify amplitude, period, phase shift, and vertical shift from a sine or cosine equation, and locate the key points of one full cycle.

Common stumbles: Looking for the period in the wrong part of the equation; forgetting to divide 2*pi by the coefficient of x rather than multiplying.

See it visually

Interactive, spoken-aloud tools that show this idea in motion.

Unit CircleAngles, radians, and sine & cosine come alive.try →
Homework for this lessonType it, write it with a stylus, or print it for pencil & paper.open →
Records attendance + 15 min of Math for this learner.
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