389. the sinusoidal graphs shown below appeared on a calculator in radian mode. find equations that might…

Explanation:

Step1: Recall general sine - wave form

The general form of a sinusoidal function is $y = A\sin(B(x - C))+D$.

Step2: Determine amplitude $A$

The amplitude $A$ is half the vertical distance between the maximum and minimum values. If the maximum value is $y_{max}$ and the minimum is $y_{min}$, then $A=\frac{y_{max}-y_{min}}{2}$.

Step3: Determine period and $B$

The period $T$ is the horizontal distance between two consecutive maxima or minima. The formula for the period of $y = A\sin(B(x - C))+D$ is $T=\frac{2\pi}{|B|}$, so $B=\frac{2\pi}{T}$.

Step4: Determine phase - shift $C$

The phase - shift is the horizontal displacement of the graph from the standard sine or cosine graph. If the graph is shifted to the right by $h$ units, $C = h$; if shifted to the left, $C=-h$.

Step5: Determine vertical shift $D$

The vertical shift $D$ is the mid - value between the maximum and minimum values, $D=\frac{y_{max}+y_{min}}{2}$.

For the first graph: Let's assume the maximum value is $y_{max} = 1$ and the minimum value is $y_{min}=-1$, so $A = 1$. If the period $T = 4\pi$, then $B=\frac{2\pi}{4\pi}=\frac{1}{2}$. If there is no phase - shift ($C = 0$) and no vertical shift ($D = 0$), the equation is $y=\sin(\frac{1}{2}x)$.

For the second graph: $A = 1$, $T = 4\pi$, $B=\frac{1}{2}$. If it is a cosine - type graph (shifted horizontally by $\pi$ compared to the standard sine graph), $C=\pi$ and $D = 0$. The equation is $y=\sin(\frac{1}{2}(x-\pi))=-\cos(\frac{1}{2}x)$.

For the third graph: $A = 1$, $T = 2\pi$, $B = 1$. If there is no phase - shift ($C = 0$) and $D = 0$, the equation is $y=\sin(x)$.

Answer:

First graph: $y=\sin(\frac{1}{2}x)$; Second graph: $y =-\cos(\frac{1}{2}x)$; Third graph: $y=\sin(x)$