To find a sinusoidal function that matches the given graph, we need to determine the amplitude, period, phase shift, and vertical shift of the sinusoidal function.

The general form of a sinusoidal function is:

$f(x) = A \sin(B(x - C)) + D$

Where:

• $A$ is the amplitude (the peak value),
• $B$ relates to the period $T$ by the formula $B = \frac{2\pi}{T}$,
• $C$ is the phase shift (horizontal shift),
• $D$ is the vertical shift.

Step 1: Determine the amplitude $A$

The amplitude is the vertical distance from the midline to a peak (or trough). From the graph, the curve seems to oscillate around a central value (the midline), but since this value isn’t explicitly given, we'll analyze the vertical distance between a peak and the nearest trough.

Step 2: Determine the period $T$

The period is the horizontal length of one complete cycle. The graph shows that the function repeats its pattern every 8 units (from $x = 0$ to $x = 8$).

Therefore, $T = 8$, and:

$B = \frac{2\pi}{T} = \frac{2\pi}{8} = \frac{\pi}{4}$

Step 3: Determine the phase shift $C$

The graph appears to start from a peak at $x = 0$. Since the standard sine function $y = \sin(x)$ starts from 0, we need to shift the sine function horizontally so that it starts from a peak.

Given that a sine function reaches its peak at $x = 0$ with no phase shift, $C = 0$.

Step 4: Determine the vertical shift $D$

The graph oscillates symmetrically around $y = 3$. Therefore, the vertical shift $D = 3$.

Step 5: Write the function

Given these parameters:

$f(x) = A \sin\left(\frac{\pi}{4}(x)\right) + 3$

To find $A$, note that the maximum and minimum values of the graph appear to be equally spaced above and below the midline $y = 3$. We need the peak and trough values to estimate $A$.

• If the peak appears to be at $y = 5$, and the trough at $y = 1$, then $A = 2$ because the amplitude is the distance from the midline to either the peak or the trough.

Thus, the sinusoidal function that fits the graph is:

$f(x) = 2 \sin\left(\frac{\pi}{4}(x)\right) + 3$

Would you like further details on any of these steps, or do you have any additional questions?

Questions:

1. How is the period of a sinusoidal function related to the parameter $B$ in the equation?
2. What role does the amplitude $A$ play in the graph of a sinusoidal function?
3. How can you determine the vertical shift $D$ from the graph of a sinusoidal function?
4. Why is it important to determine the phase shift $C$ when matching a sinusoidal graph?
5. How would the function change if the sinusoidal graph started from a trough rather than a peak?

Tip:

Remember that the sine and cosine functions are closely related—any sine function can be written as a cosine function with an appropriate phase shift and vice versa.