Identifying the Equation for a Sinusoidal Function from a Graph

To determine the equation, we need to identify each value in the general form of a sinusoidal function.

The graph could represent either a sine or a cosine function that is shifted and/or reflected. When $x=0,$ the graph has an extreme point, $\left(0,0\right).$ Since the cosine function has an extreme point for $x=0,$ let us write our equation in terms of a cosine function.

Let’s start with the midline. We can see that the graph rises and falls an equal distance above and below $y=\mathrm{0.5.}$ This value, which is the midline, is $D$ in the equation, so $D=\mathrm{0.5.}$

The greatest distance above and below the midline is the amplitude. The maxima are 0.5 units above the midline and the minima are 0.5 units below the midline. So $\left|A\right|=\mathrm{0.5.}$ Another way we could have determined the amplitude is by recognizing that the difference between the height of local maxima and minima is 1, so $\left|A\right|=\frac{1}{2}=\mathrm{0.5.}$ Also, the graph is reflected about the x-axis so that $A=-\mathrm{0.5.}$

The graph is not horizontally stretched or compressed, so $B=\mathrm{1;}$ and the graph is not shifted horizontally, so $C=0.$

Putting this all together,