The problem involves determining the function that represents the dotted graph, based on the transformations applied to the original function $y = f(x)$, represented by the solid black graph.

Analysis:

The dotted graph appears to be a horizontally shifted and reflected version of the original graph.

1. Original function ($y = f(x)$):

   • The black graph resembles the quadratic function $y = x^2$, but we don't explicitly need the form of $f(x)$. It's sufficient to compare the transformations between the solid and dotted graphs.

2. Transformation applied:

   • The dotted graph has been reflected horizontally across the $y$-axis.
   • It has also been shifted horizontally to the left by 3 units.

Transformed function:

A horizontal reflection involves replacing $x$ with $-x$. A horizontal shift to the left by 3 involves replacing $x$ with $x + 3$. Combining these gives the transformation: $y = f(-(x + 3)).$

Thus, the function representing the dotted graph is: $y = f(-x - 3).$

Let me know if you need further details! Here are some follow-up questions to deepen understanding:

Relative Questions:

1. What does a horizontal reflection do to the graph of any function?
2. How does a shift of $x$ to the left differ from a shift to the right?
3. Can you derive the general rule for combining multiple transformations of functions?
4. How does replacing $x$ with $-x$ affect the symmetry of the graph?
5. Could vertical transformations also apply to this graph? How would they look?

Tip:

Always apply transformations step-by-step and in the correct order: reflections, scaling, shifts. This avoids errors when combining multiple transformations.