https://file.aiquickdraw.com/mathbot/file/1729608949685r9u16ram.jpg

Let's analyze the function:

\[
y = \frac{3}{2} \sqrt{x + 4} - 3
\]

### Step 1: Identify key features of the function
- **Domain**: The square root function \(\sqrt{x + 4}\) requires that \(x + 4 \geq 0\), so \(x \geq -4\). Therefore, the domain of the function is \(x \geq -4\).
- **Range**: Since the square root function outputs non-negative values, and the coefficient of \(\frac{3}{2}\) stretches the graph, the lowest value will occur when \(x = -4\). At \(x = -4\), \(\sqrt{x + 4} = 0\), so the lowest \(y\)-value will be:

  \[
  y = \frac{3}{2}(0) - 3 = -3
  \]
  As \(x\) increases, the function increases because of the positive coefficient \(\frac{3}{2}\). Therefore, the range is \(y \geq -3\).

- **Vertical shift**: The function is shifted 3 units downward because of the \(-3\) term.
- **Horizontal shift**: The function is shifted 4 units to the left because of the \(x + 4\) under the square root.

### Step 2: Determine the correct graph
- The graph should start at \(x = -4\) and have a \(y\)-value of \(-3\), and it should increase slowly as \(x\) increases.
- The graph should have a slight curve upwards, since it's based on the square root function, and should only exist for \(x \geq -4\).

### Step 3: Compare with the provided options
Looking at the graphs provided:

- **Graph A**: This graph curves downward, which does not match the behavior of the function (should increase).
- **Graph B**: This graph starts at \(x = -4\) and \(y = -3\), then increases, which matches the description.
- **Graph C**: This graph starts higher than \(-3\) and appears incorrect.
- **Graph D**: Similar to C, it starts at an incorrect height.

### Conclusion:
The correct graph is **B**.

Do you want further clarification or details? Here are five follow-up questions to deepen your understanding:

1. How do you determine the domain of a square root function in general?
2. How does the vertical shift affect the range of a function?
3. What role does the coefficient \(\frac{3}{2}\) play in stretching or compressing the graph?
4. How do you graphically interpret a horizontal shift in functions?
5. Why does the square root function only exist for certain \(x\)-values?

**Tip**: Always check both vertical and horizontal shifts when graphing transformations of functions!

Graph the function y = (3/2) * sqrt(x + 4) - 3 and choose the correct graph from the given options.

Analyze and correctly graph the function y = (3/2) * sqrt(x + 4) - 3. This problem involves transformations of square root functions, including vertical and horizontal shifts. Learn to determine the domain, range, and behavior of the graph step-by-step. Suitable for students in Grades 9-11.

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