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All courses Math III · F-TF.2.1 35 of 55
Graph all six basic trigonometric functions.

Specify complete sine graph feature sets on an interval

Problem
Specify the complete key-feature data for y=sin(x) on [0,2pi]. Enter domain interval, number of complete cycles, amplitude, period, midline, range, and ordered sets of all zeros, maxima, and minima in the interval (including allowed endpoints).
Exact sin curve on [0,2π] with radian axes and all key locations visible but unlabeled. Open full size
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Hint

List multiples of each feature formula that lie in the stated interval, checking endpoints.

For y=sin x, amplitude=1, period=2pi, midline y=0, zeros at kpi, maxima at pi/2+2kpi, and minima at 3pi/2+2kpi.

Solution walkthrough

01

Record global features

\[\text{amplitude}=1;~\text{period}=2\text{pi};~\text{midline}~y=0;~\text{range}~[-1,1]\]

These features are fixed for the parent sine function.

02

Count complete cycles

\[\text{domain}~[0,2\text{pi}];~\text{complete}~\text{cycles}=1\]

Divide interval length by 2pi and respect closed endpoints.

03

List midline zeros

\[{(0,0),(\text{pi},0),(2\text{pi},0)}\]

Keep every x=kpi in the interval.

04

List extrema

\[\text{maxima}={(\text{pi}/2,1)};~\text{minima}={(3\text{pi}/2,-1)}\]

Use the positive and negative quarter-period anchors for every cycle.

05

Submit complete graph data

\[\text{domain}=[0,2\text{pi}];~\text{complete}~\text{cycles}=1;~\text{amplitude}=1;~\text{period}=2\text{pi};~\text{midline}=y=0;~\text{range}=[-1,1];~\text{zeros}={(0,0),(\text{pi},0),(2\text{pi},0)};~\text{maxima}={(\text{pi}/2,1)};~\text{minima}={(3\text{pi}/2,-1)}\]

These exact anchors determine an accurate sketch.

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Another way

  1. Build a quarter-period table with x-spacing pi/2 and repeat the values 0,1,0,-1,0.

!

Common mistake

Do not omit repeated-cycle points or closed endpoints, and do not interchange domain and range.

Exact sin curve on [0,2π] with zeros, extrema, midline, period, and range identified.
zeros x=0,π,2π; max (π/2,1); min (3π/2,−1) period 2π; midline y=0; range [−1,1]