To find a trigonometric function that matches the graph in the form $y = A \sin(kx) + C$ or $y = A \cos(kx) + C$, we can break down the components based on the properties of the graph.

Key properties to observe:

1. Amplitude (A): The graph oscillates between 5 and -5, which means the amplitude is 5. So, $A = 5$.

2. Vertical Shift (C): The midline of the graph is at $y = 0$, so there is no vertical shift. Thus, $C = 0$.

3. Period (T): To find the period, observe the x-axis. One full cycle of the graph (from peak to peak) spans from $x = -12$ to $x = 0$, so the period $T$ is 12. The relationship between period $T$ and $k$ is given by the formula $T = \frac{2\pi}{k}$. Solving for $k$, we get: $T = 12 \implies \frac{2\pi}{k} = 12 \implies k = \frac{2\pi}{12} = \frac{\pi}{6}.$

4. Phase Shift: Since the graph looks like a cosine function starting at a peak at $x = -12$, we will use the cosine form of the function $y = A \cos(kx) + C$ with no phase shift required.

The resulting function:

$y = 5 \cos\left(\frac{\pi}{6} x\right)$

Would you like any further clarification or details on this process?

Here are five related questions:

1. How would the function change if there were a phase shift?
2. Can we model this graph using a sine function instead of a cosine function?
3. How would the function change if the amplitude were doubled?
4. What would happen to the period if $k$ were doubled?
5. How do vertical shifts affect the graph of trigonometric functions?

Tip: When analyzing trigonometric graphs, always check the amplitude, period, and any shifts to determine the equation correctly.