Question

find the vertical shift of the sinusoidal function.
simplify any fractions.
vertical shift =

Explanation:

Step1: Recall vertical shift formula

The vertical shift of a sinusoidal function is the midline, calculated as $\frac{\text{maximum value} + \text{minimum value}}{2}$.

Step2: Identify max and min values

From the graph, the maximum value (peak) is $7$? Wait, no, looking at the y - axis, the peaks seem to be at y = 7? Wait, no, the graph's peaks and troughs: the troughs (minimums) are at y = 1, and the peaks (maximums) are at y = 7? Wait, no, let's check the grid. Wait, the y - axis has markings: the troughs are at y = 1, and the peaks: let's see the distance. Wait, maybe I misread. Wait, the vertical shift is the midline, so midline is average of max and min. Let's find max and min. Looking at the graph, the minimum value (lowest point of the trough) is $1$, and the maximum value (highest point of the peak) is $7$? Wait, no, wait the y - axis: the troughs are at y = 1, and the peaks: let's check the distance between peak and trough. The amplitude is (max - min)/2, and vertical shift is (max + min)/2. Wait, from the graph, the troughs are at y = 1, and the peaks: let's see, the vertical distance between peak and trough: from y = 1 to y = 7? Wait, no, maybe the peak is at y = 7? Wait, no, the y - axis has 0,1,2,3,4,5,6,7. Wait, the troughs are at y = 1, and the peaks: let's check the graph. Wait, the function's midline (vertical shift) is the average of the maximum and minimum values. Let's find max and min. The minimum value (min) is 1, the maximum value (max) is 7? Wait, no, wait the graph: the troughs are at y = 1, and the peaks: let's see, the distance from trough to peak: from y = 1 to y = 7, so the midline is (1 + 7)/2 = 4? Wait, no, wait maybe I made a mistake. Wait, looking at the graph again, the troughs are at y = 1, and the peaks: let's check the y - axis. Wait, the vertical shift is the midline, so if the minimum is 1 and maximum is 7, then vertical shift is (1 + 7)/2 = 4. Wait, but let's confirm. Alternatively, maybe the minimum is 1 and maximum is 7, so midline (vertical shift) is (1 + 7)/2 = 4. Wait, but let's check the graph. The graph oscillates around the midline. So if the troughs are at y = 1 and peaks at y = 7, then midline is 4.
Wait, maybe I misread the max and min. Wait, the y - axis: the troughs are at y = 1, and the peaks: let's see the grid. Each square is 1 unit? So the troughs are at y = 1, and the peaks: let's see, the distance from y = 1 to y = 7 is 6 units, so amplitude is 3, and vertical shift is 1+3 = 4 (or 7 - 3 = 4). So vertical shift is 4.

Step1: Determine max and min values

From the graph of the sinusoidal function, the minimum value (minimum of the troughs) is $y_{min}=1$ and the maximum value (maximum of the peaks) is $y_{max}=7$.

Step2: Calculate vertical shift

The formula for the vertical shift (midline) of a sinusoidal function is $V=\frac{y_{max}+y_{min}}{2}$.

Substitute $y_{max} = 7$ and $y_{min}=1$ into the formula:

$V=\frac{7 + 1}{2}=\frac{8}{2}=4$

Answer:

4