sformations on the basic functions to write a rule y = f (x) that would produce the given graph.

To write a rule $y = f(x)$ that matches a given graph, follow these general steps:

1. ### Analyze the Graph

• Identify key features of the graph such as intercepts, asymptotes, maxima, minima, and intervals of increase or decrease.

2. ### Determine the Basic Function

• Decide on the type of basic function that may fit the graph. Common basic functions include linear, quadratic, cubic, exponential, logarithmic, and trigonometric functions.

3. ### Identify Transformations

• Consider transformations such as:
  • Vertical shifts: Moving the graph up/down ($y = f(x) + k$).
  • Horizontal shifts: Moving the graph left/right ($y = f(x - h)$).
  • Stretching or compressing: Vertically or horizontally ($y = af(x)$ or $y = f(bx)$).
  • Reflections: Flipping the graph over the x-axis or y-axis ($y = -f(x)$ or $y = f(-x)$).

4. ### Construct the Function

• Combine the selected basic function with the identified transformations to formulate the function rule $y = f(x)$.

Final Answer

Present the final rule $y = f(x)$ based on your analysis of the graph and the identified transformations. For example, if after your analysis you find the function should be a vertically stretched quadratic and shifted 2 units up, your rule might look like:

$y = 2(x - h)^2 + k$

where $h$ and $k$ will depend on the graph's features.

Please provide more specific details about the graph for a tailored function!