https://file.aiquickdraw.com/mathbot/file/1726876241425xw31tyx8.jpg

The graph provided in the image appears to be a sinusoidal wave. From the general shape of the function and the grid, it seems that this is a trigonometric function, likely of the form:

\[
f(x) = A \sin(Bx + C) + D \quad \text{or} \quad f(x) = A \cos(Bx + C) + D
\]

Let's analyze the key features of the graph to determine the equation:

### 1. **Amplitude (A):**
   - The maximum value of the function is 3, and the minimum value is -3, so the amplitude is \( A = 3 \).

### 2. **Period:**
   - The graph appears to complete one full cycle between 0 and \( 2\pi \). The period \( T \) is \( 2\pi \). For a sine or cosine function, the period is given by \( \frac{2\pi}{B} \). In this case, \( B = 1 \).

### 3. **Horizontal Shift (C):**
   - The function has not been shifted horizontally, so \( C = 0 \).

### 4. **Vertical Shift (D):**
   - There is no vertical shift in the graph because the function oscillates symmetrically about the x-axis, meaning \( D = 0 \).

### Function Type (Cosine):
   - The graph appears to resemble a cosine function starting at a maximum point (3 at \(x = 0\)) rather than starting from 0, which is typical of a cosine function. So, the function is more likely to be:

\[
f(x) = 3 \cos(x)
\]

### Conclusion:
The rule representing this function is:

\[
f(x) = 3 \cos(x)
\]

Would you like further details on this solution or have any questions?

### Related Questions:
1. How does changing the amplitude \(A\) affect the graph of a trigonometric function?
2. What is the effect of a horizontal shift \(C\) on the graph of sine or cosine functions?
3. How can we derive the period of any trigonometric function given its equation?
4. What happens if a vertical shift \(D\) is introduced to the function?
5. Can you describe the graph of \( f(x) = \sin(x) \) and how it compares to \( f(x) = \cos(x) \)?

### Tip:
When identifying the rule of a trigonometric graph, always check for key characteristics like amplitude, period, phase shift, and vertical shift. This helps in constructing the exact equation.

Jan is doing research on the phenomena of vibrations. She compiles a series of results and obtains the following graph on the computer screen. What rule represents this function?

Explore the representation of a cosine function based on graph analysis. This problem covers essential trigonometric concepts such as amplitude, period, and cosine function structure. Suitable for Grades 10-12 students.

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