To write the equation of this trigonometric graph, let's analyze its properties:

1. Amplitude: The graph ranges from -5 to -1 on the $y$-axis, so the amplitude is the distance from the midline to the peak (or trough). The midline here is $y = -3$, and the amplitude is $|5 - 3| = 2$.

2. Vertical Shift: The graph is centered around $y = -3$ rather than $y = 0$, so there is a vertical shift down by 3 units.

3. Period: The graph completes a full cycle from $-\pi$ to $\pi$, which gives a period of $2\pi$.

4. Equation Form: The graph resembles a cosine function with a reflection across the $y$-axis (it starts at a minimum point). Thus, the function form should be: $y = -a \cos(bx) + d$ where $a$ is the amplitude, $b$ affects the period, and $d$ is the vertical shift.

Given these observations:

• The amplitude $a = 2$,
• The period $2\pi$ indicates $b = 1$,
• The vertical shift $d = -3$.

Equation

Thus, the equation of the graph is: $y = -2 \cos(x) - 3$

Would you like more details, or do you have any questions?

Here are five related questions you might find interesting:

1. How would the equation change if the amplitude increased?
2. What is the effect of altering the vertical shift in a trigonometric function?
3. How can we determine if a trigonometric function is a sine or cosine function from its graph?
4. What changes would be necessary if the graph had a different period?
5. How can you determine if a function should be positive or negative based on the graph?

Tip: When determining a trigonometric function's equation, always identify amplitude, period, and vertical shifts first—they provide the fundamental shape and position of the graph.