The given graph exhibits a repeating wave pattern, and we need to determine its equation in the form:

$y = a \sin(bx) \quad \text{or} \quad y = a \cos(bx)$

Step 1: Identify the Amplitude

The amplitude $a$ is the absolute value of the maximum or minimum y-values. The given graph oscillates between 4 and -4, so:

$a = 4$

Step 2: Identify the Function Type (Sine or Cosine)

• The given graph starts at $(0,4)$, which corresponds to the maximum value.
• The cosine function has a standard form $y = a \cos(bx)$ that starts at a maximum when $a > 0$.
• Since this graph follows that pattern, the function is cosine-based.

Step 3: Determine $b$ Using the Period

The standard cosine function has a period of $\frac{2\pi}{b}$.
From the given graph, one full cycle occurs from $x = 0$ to $x = \frac{\pi}{2}$. So, the period is:

$\frac{2\pi}{b} = \frac{\pi}{2}$

Solving for $b$:

$b = \frac{2\pi}{\frac{\pi}{2}} = 4$

Step 4: Write the Final Equation

$y = 4 \cos(4x)$

Let me know if you need more details!

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Related Questions:

1. How would the equation change if the function were a sine function instead?
2. What effect does changing $a$ have on the graph?
3. How does the value of $b$ affect the frequency of the function?
4. What would be the equation if the graph were reflected across the x-axis?
5. How can you determine the equation if given only a table of values?

Tip:

The coefficient $b$ determines the number of cycles per $2\pi$. A larger $b$ means more oscillations in the same interval! 🚀