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All courses Algebra II · F-TF.2.1 42 of 55
Graph all six basic trigonometric functions.

Specify complete sine graph feature sets on an interval

Problem
Use the graph of \(y~=~\sin(x)\) on \([0,~2\pi~]\) to state the domain, complete-cycle count, amplitude, period, midline, range, and all zeros, maxima, and minima on the interval.
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

Read the interval and cycle

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

The graph is restricted to the closed interval from 0 through 2pi, whose length is one sine period.

02

Read the vertical structure

\[\text{maximum}~y=1;~\text{minimum}~y=-1;~\text{midline}~y=0\]

The curve is centered on y=0 and reaches one unit above and below it, so the amplitude is 1 and the range is [-1,1].

03

Determine the period

\[\text{period}=2\pi\]

The pattern beginning at (0,0) returns to the same position and direction at (2pi,0), so one full cycle spans 2pi.

04

List all featured points

\[\text{zeros}:~(0,0),(\pi,0),(2\pi,0);~\text{maxima}:~(\pi/2,1);~\text{minima}:~(3\pi/2,-1)\]

The zeros are every x-axis crossing on the closed interval; the highest and lowest marked points give the sole maximum and minimum.

05

State the complete graph description

\[\text{The}~\text{domain}~\text{is}~[0,~2\pi];~\text{there}~\text{are}~1~\text{complete}~\text{cycle}(s),~\text{amplitude}~1,~\text{period}~2\pi,~\text{midline}~y~=~0,~\text{and}~\text{range}~[-1,~1].~\text{Zeros}~\text{are}~(0,0),~(\pi,0),~(2\pi,0);~\text{maxima}~\text{are}~(\pi/2,1);~\text{minima}~\text{are}~(3\pi/2,-1).\]

This combines the horizontal, vertical, and point features read from the entire displayed interval.

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Another way

  1. Generate the five key sine points over one period from the unit circle, then classify their coordinates and infer the continuous range.

!

Common mistake

Do not omit the endpoint zero at 2pi. The displayed domain is closed, so both 0 and 2pi are included.

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]

Solution walkthrough video