For the prescribed families y=asin(bx)+c, y=acos(bx)+c, y=atan(bx)+c, a is a positive integer, b 0 is a simple fraction or integer, and c is an integer. Here positive b is used for the basic reconstruction cases. A negative b reflects the base graph in the y-axis as well as changing its horizontal scale by 1/|b|. A point (u,v) on the base graph maps to (u/b,av+c). Therefore one sine/cosine turn takes 2pi/|b| radians or 360 /|b|; tangent takes pi/|b| or 180 /|b|. For sine/cosine, A=(y_max-y_min)/2=a, c=(y_max+y_min)/2. Read the least positive period, then find b. For tangent there is no amplitude because there are no highest or lowest values. For y=2sin(3x)-1, the horizontal scale factor is 1/3, vertical scale factor is 2, midline is y=-1, amplitude is 2 and period is 2pi/3. The base point (pi/2,1) maps to (pi/6,1). Do not call the maximum the amplitude, or assume the displayed interval is one period. A restricted portion of a sine curve may never reach c+a. In a reconstruction, use the least positive period, not the gap between arbitrary equal heights.

For the prescribed families \(y=a\sin(bx)+c\), \(y=a\cos(bx)+c\), \(y=a\tan(bx)+c\), \(a\) is a positive integer, \(b\ne0\) is a simple fraction or integer, and \(c\) is an integer. Here positive b is used for the basic reconstruction cases.

A negative b reflects the base graph in the y-axis as well as changing its horizontal scale by \(\frac{1}{|b|}\). A point \((u,v)\) on the base graph maps to \(\left(\frac{u}{b},av+c\right)\). Therefore one sine/cosine turn takes \(\frac{2\pi}{|b|}\) radians or \(\frac{360^\circ}{|b|}\); tangent takes \(\frac{\pi}{|b|}\) or \(\frac{180^\circ}{|b|}\).

For sine/cosine, \(A=\frac{y_{\max}-y_{\min}}{2}=a\), \(c=\frac{y_{\max}+y_{\min}}{2}\). Read the least positive period, then find b. For tangent there is no amplitude because there are no highest or lowest values.

For \(y=2\sin(3x)-1\), the horizontal scale factor is \(\frac{1}{3}\), vertical scale factor is 2, midline is \(y=-1\), amplitude is 2 and period is \(\frac{2\pi}{3}\). The base point \(\left(\frac{\pi}{2},1\right)\) maps to \(\left(\frac{\pi}{6},1\right)\).

Do not call the maximum the amplitude, or assume the displayed interval is one period. A restricted portion of a sine curve may never reach \(c+a\). In a reconstruction, use the least positive period, not the gap between arbitrary equal heights.

Amplitude period and related graphs

Amplitude period and related graphs