Eleventh Grade · Math · week 35

Graphing sine and cosine

Graph sine and cosine functions and identify amplitude, period, and vertical shift.

The graphs of y=sin(x) and y=cos(x) are smooth, repeating waves between -1 and 1, built directly from the unit circle values you just learned -- literally plot (angle, sin(angle)) or (angle, cos(angle)) as the angle sweeps around the circle, and the wave appears. Both complete one full cycle every 2*pi and never leave the range [-1, 1] in their basic form.

Three numbers describe any transformed sine or cosine wave: amplitude (the height of the wave above and below its center -- the coefficient out front, in absolute value), period (the length of one complete cycle -- 2*pi divided by the coefficient of x inside), and vertical shift (how far up or down the whole wave is centered, added or subtracted outside). A negative sign out front flips the wave upside down.

📋 For the teacher
Teach it (10–15 min):
  • Build the sine wave together by literally plotting the unit-circle sine values at 0, pi/2, pi, 3pi/2, and 2pi and connecting them smoothly -- the shape should feel earned, not assumed.
  • Isolate amplitude and period as two completely separate questions with two completely separate locations in the equation: amplitude lives OUTSIDE (multiplying the whole function), period comes from INSIDE (attached to x).
  • Have them predict a graph's shape from its equation BEFORE plotting a single point, then check the prediction -- building the muscle of reading equations, not just plotting them.
  • Compare several period changes side by side (coefficient 1, 2, and 1/2 on x) so "bigger coefficient means faster wave, smaller period" becomes visually obvious.
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?
  • What does a vertical shift do to the center line of a sine or cosine graph?

Mastery looks like: They correctly identify amplitude, period, and vertical shift from a sine or cosine equation and can evaluate the function at the standard key points.

Common stumbles: Looking for the period in the wrong part of the equation (outside instead of inside, or vice versa); 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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