Answer:
y = cos (x + pi/6)
Step-by-step explanation:
We are told this is a translation. Look at the original graph which is dashed. That is the graph of y = cos x. Now look at the translated graph which is solid. Compared to the dashed graph, the solid line is translated to the left.
The question now is how much of a horizontal translation it has. The scale on the x axis can be determined. The period of the cosine function is 2pi. At x = 0, the graph of y = cos x has a maximum value. At x = 2pi, the graph has a maximum value again. From the highest point of he dashed graph to the next highest point, the distance along the x-axis must be 2pi. That means the second maximum we see at the right is at x = 2pi. Since the 2pi interval is divided into 12 equal spaces, that means each vertical line along the x-axis represents 2pi/12, or pi/6. Notice that the solid graph is pi/6 to the left of the dashed line, so the translation is pi/6 to the left, or -pi/6.
A horizontal translation is obtained when x is replaced by x - h.
cos x is the original function.
cos (x - h) is the translated function, where h is the translation.
Since the translation is -pi/6, then h = -pi/6
cos [x - (-pi/6)] = cos (x + pi/6)
G, the translated function has equation
y = cos (x + pi/6)
