Identifying the Equation for a Sinusoidal Function from a Graph

With the highest value at 1 and the lowest value at $\mathrm{-5},$ the midline will be halfway between at $\mathrm{-2.}$ So $D=\mathrm{-2.}$

The distance from the midline to the highest or lowest value gives an amplitude of $\left|A\right|=3.$

The period of the graph is 6, which can be measured from the peak at $x=1$ to the next peak at $x=7,$ or from the distance between the lowest points. Therefore, $P=\frac{2\pi }{\left|B\right|}=6.$ Using the positive value for $B,$ we find that

So far, our equation is either $y=3\sin \left(\frac{\pi }{3}x-C\right)-2$ or $y=3\cos \left(\frac{\pi }{3}x-C\right)-2.$ For the shape and shift, we have more than one option. We could write this as any one of the following:

• a cosine shifted to the right
• a negative cosine shifted to the left
• a sine shifted to the left
• a negative sine shifted to the right

While any of these would be correct, the cosine shifts are easier to work with than the sine shifts in this case because they involve integer values. So our function becomes

Again, these functions are equivalent, so both yield the same graph.