Solution:

The equation of a sinusoidal function generally takes the form:

$f(x) = A \sin(B(x - C)) + D \quad \text{or} \quad f(x) = A \cos(B(x - C)) + D$

Where:

• $A$: Amplitude
• $B$: Angular frequency
• $C$: Horizontal shift (phase shift)
• $D$: Vertical shift

Step 1: Determine the amplitude ($A$)

From the graph:

• Maximum value: $5$
• Minimum value: $-5$
• Amplitude: $A = \frac{\text{Maximum} - \text{Minimum}}{2} = \frac{5 - (-5)}{2} = 5$

Step 2: Determine the vertical shift ($D$)

• Vertical shift ($D$) is the midpoint between the maximum and minimum values: $D = \frac{\text{Maximum} + \text{Minimum}}{2} = \frac{5 + (-5)}{2} = 0$

Step 3: Determine the period

The period ($T$) can be calculated using the distance between repeating points. From the graph, the function repeats over a length of $10$ units (e.g., from $x = -9$ to $x = 1$).

• Period: $T = 10$

The angular frequency ($B$) is related to the period by: $B = \frac{2\pi}{T} = \frac{2\pi}{10} = \frac{\pi}{5}$

Step 4: Determine the horizontal shift ($C$)

The graph passes through $(-9, -4)$ and $(1, -4)$. The phase shift depends on whether we use the sine or cosine form:

1. Cosine form: The cosine function starts at its maximum or minimum. From the graph, the cosine is shifted to the left. The horizontal shift is $-9$.

   $C = -9$

2. Sine form: The sine function starts at the midpoint (where the function crosses the vertical shift line). From the graph, the sine crosses at $x = -7$.

   $C = -7$

Step 5: Write the equation

1. Using cosine: $f(x) = 5 \cos\left(\frac{\pi}{5}(x + 9)\right)$

2. Using sine: $f(x) = 5 \sin\left(\frac{\pi}{5}(x + 7)\right)$

Do you want further clarification on these steps? Let me know if you'd like to explore any specific part!

Related Questions:

1. How do we find the phase shift for general sinusoidal graphs?
2. How can you convert between sine and cosine forms of the same graph?
3. What is the effect of amplitude and vertical shift on sinusoidal graphs?
4. How do you determine the angular frequency when the period is a fraction?
5. Why does adding or subtracting a phase shift affect the graph’s starting point?

Tip: Always verify your calculated period by testing points on the graph to ensure the function repeats correctly!