Amplitude, Period, and Phase Shift

Reading the Parameters of a Sinusoid

A sine or cosine graph repeats, but its equation can change how tall the graph is, how quickly it repeats, and where a cycle begins along the horizontal axis. In the forms considered here, those features are encoded directly by three parameters.

Here \(A\), \(B\), and \(C\) are real numbers. The coefficient \(A\) scales the output, \(B\) scales the input angle, and \(C\) specifies a horizontal displacement. Reading these parameters carefully gives a reliable way to plot one complete cycle and continue the repeating curve.

Amplitude, Period, and Phase Shift

The sine and cosine functions have values between \(-1\) and \(1\). Multiplying either function by \(A\) makes its output range from \(-|A|\) to \(|A|\). Thus the amplitude is a nonnegative distance from the midline \(y=0\) to an extreme value. It is not necessarily the signed coefficient \(A\): for example, if \(A=-3\), the amplitude is \(3\).

The period describes the horizontal distance for one full repetition. The input to sine or cosine is \(B(x-C)\), so changing \(x\) by \(T\) changes that input by \(BT\). One full cycle occurs when the input changes by \(2\pi\) in magnitude. Therefore \(T=2\pi/|B|\). The absolute value matters: a period is positive even if \(B\) is negative.

The phase shift is read from the grouped expression \(x-C\). If \(C>0\), the shift is \(C\) units to the right; if \(C<0\), it is \(|C|\) units to the left. For instance, \(x+\frac{\pi}{4}\) can be written \(x-(-\frac{\pi}{4})\), so its shift is \(-\frac{\pi}{4}\), or left by \(\frac{\pi}{4}\). The location \(x=C\) is where the argument \(B(x-C)\) equals zero.

These parameter readings describe the equation as written. Since the graph repeats every \(T\), shifting \(C\) by an integer multiple of \(T\) gives an equivalent horizontal placement of the same repeating graph.

Why the Period Formula Works

For either sine or cosine, adding \(2\pi\) or subtracting \(2\pi\) to the angle leaves the function value unchanged. If \(T=2\pi/|B|\), then \(BT=2\pi\) when \(B>0\) and \(BT=-2\pi\) when \(B<0\). Consequently, increasing \(x\) by \(T\) changes the angle by one full cycle, in either direction.

where \(\operatorname{sgn}(B)=1\) for \(B>0\) and \(\operatorname{sgn}(B)=-1\) for \(B<0\). Periodicity of sine and cosine then gives the same output at \(x+T\) as at \(x\). The earlier material on periodicity establishes that their fundamental positive period in the angle is \(2\pi\). Thus the smallest positive horizontal period is \(2\pi/|B|\).

If \(A=0\), the equation is the constant function \(y=0\); if \(B=0\), it is constant as well. A constant function repeats for every positive distance, so it does not have a unique fundamental period. In those cases, do not use the period formula to claim a unique cycle length.

Five Key Points for One Cycle

For a nonconstant graph, let \(T=2\pi/|B|\). The five \(x\)-coordinates from \(C\) through one full cycle are \(C,C+T/4,C+T/2,C+3T/4,C+T\). The spacing is \(T/4\), and the points are in increasing \(x\)-order. Their output values depend on whether the function is sine or cosine and, for sine, on the sign of \(B\).

To see the sine entries, substitute \(x=C+jT/4\), where \(j=0,1,2,3,4\). The angle becomes \(B jT/4=j\pi\,\operatorname{sgn}(B)/2\). Evaluating sine at these angles gives the displayed values, including the reversal of the middle two nonzero values when \(B<0\). For cosine, changing the sign of an angle does not change its cosine, so the listed sequence is the same for either sign of \(B\).

Connect the plotted points with a smooth sinusoidal curve and continue the pattern every \(T\) units. The table gives the points, not straight-line segments: sine and cosine graphs are smooth curves.

Worked Examples

A Common Mistake: Ignoring the Sign of \(B\)

A frequent error is to use the usual increasing-angle sine pattern even when \(B\) is negative. Consider \(y=\sin(-x)\), with \(A=1\), \(B=-1\), \(C=0\), and \(T=2\pi\). The five coordinates are \(0,\frac{\pi}{2},\pi,\frac{3\pi}{2},2\pi\). The correct heights are \(0,-1,0,1,0\), because the angle is \(-x\). Using the positive-\(B\) pattern would incorrectly give \(0,1,0,-1,0\). Check the second and fourth values directly: \(\sin(-\frac{\pi}{2})=-1\), while \(\sin(-\frac{3\pi}{2})=1\). Use the sign-aware table whenever you sketch a sine graph.

Another useful check is to verify the units of your calculation in the horizontal direction: the period must be positive, and the five key points must be separated by exactly \(T/4\). A negative result from \(2\pi/B\) signals that the absolute value in the period formula was omitted.

Why These Parameters Matter

Amplitude, period, and phase shift let you compare sinusoidal graphs without plotting many points first. Amplitude tells you the size of the vertical range, period tells you the horizontal length of a cycle, and phase shift locates the cycle. Together with the correct sine or cosine key-point pattern, they turn an equation into a dependable sketch.