From the graph, it appears we are working with a transformed logarithmic function. In general, logarithmic transformations can involve translations, reflections, and stretches/compressions.

Observations

1. The graph has a vertical asymptote at $x = -3$, indicating a horizontal shift to the left by 3 units.
2. The function is decreasing as $x$ increases, which suggests a reflection over the x-axis.
3. The behavior resembles that of a standard log function, flipped and shifted.

Equation Form

Given these transformations, the general form of the function can be written as: $f(x) = -\log(x + 3)$ This accounts for:

• The shift left by 3, represented by $(x + 3)$.
• The reflection over the x-axis, represented by the negative sign.

Final Answer

$f(x) = -\log(x + 3)$

Would you like further explanation, or do you have questions about this solution?

Here are 5 related questions to deepen understanding:

1. How would a vertical stretch affect the equation of a logarithmic function?
2. What would the graph look like if we shifted it right instead of left?
3. How does reflecting over the y-axis change the equation of a logarithmic function?
4. What are the effects of adding a constant outside the log function, like $-\log(x + 3) + 2$?
5. How do you determine the domain of transformed logarithmic functions?

Tip: When examining transformations, always start by identifying asymptotes and reflections. These provide clues to shifts and direction changes.