To solve the problem from the uploaded image, we need to identify the sinusoidal function $f(x)$ that matches the graph provided. Here's the process:

Step 1: General form of the sinusoidal function

The general form of a sinusoidal function is: $f(x) = A \sin(B(x - C)) + D$ or $f(x) = A \cos(B(x - C)) + D$ where:

• $A$ is the amplitude,
• $B$ determines the period ($\text{Period} = \frac{2\pi}{B}$),
• $C$ is the horizontal shift,
• $D$ is the vertical shift.

From the graph:

• The graph oscillates between -2 and 2, so the amplitude $A = 2$.
• The midline (vertical shift) is $D = 0$, as the graph is centered around the x-axis.
• The period can be determined from the x-values of two consecutive peaks or troughs.

Step 2: Period and $B$

From the graph:

• One full cycle occurs between $-11$ and $3$, a span of $14$ units.
• Thus, the period $P = 14$, and $B = \frac{2\pi}{P} = \frac{2\pi}{14} = \frac{\pi}{7}$.

Step 3: Phase shift ($C$)

The graph crosses the midline (0) at $x = -11$ going upward. This point indicates the start of the sine wave, so $C = -11$.

Step 4: Final equation

Now substitute the values into the sine form: $f(x) = 2 \sin\left(\frac{\pi}{7}(x + 11)\right)$

Alternatively, if you prefer a cosine function: $f(x) = 2 \cos\left(\frac{\pi}{7}(x + 18)\right)$ where $x = -18$ represents a cosine peak.

Let me know if you'd like a detailed breakdown of any step! Here are some related questions for further exploration:

1. How do you calculate the amplitude of a sinusoidal function from a graph?
2. What is the difference between a sine and cosine function graphically?
3. How do horizontal shifts affect the equation of a sinusoidal function?
4. Can a sinusoidal function have a vertical shift different from zero? How does it affect the graph?
5. How do you find the phase shift $C$ if the graph doesn’t pass through the origin?

Tip:

When working with sinusoidal graphs, always check the midline, amplitude, and period first—they form the foundation for identifying the function's equation.