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.