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Let's break down the mathematical information from the image you uploaded.

The function shown is \( f(x) = -\sqrt{x + 3} - 4 \).

### 1. **The endpoint of the graph:**
This function involves a square root, which means the graph has a starting point, as the square root function is only defined for non-negative values of the argument inside the square root.

To find the endpoint:
- The expression inside the square root must be greater than or equal to zero:
  \[
  x + 3 \geq 0 \implies x \geq -3
  \]
- So, the function is defined starting from \( x = -3 \).
- Substituting \( x = -3 \) into the function:
  \[
  f(-3) = -\sqrt{-3 + 3} - 4 = -\sqrt{0} - 4 = -4
  \]
  Therefore, the endpoint of the graph is at \( (-3, -4) \).

### 2. **The domain of the function:**
The domain is all \( x \)-values for which the function is defined, based on the condition \( x + 3 \geq 0 \), or \( x \geq -3 \).

Thus, the domain is:
\[
[-3, \infty)
\]

### 3. **The range of the function:**
The range is all possible \( y \)-values that the function can take. Since the square root function \( \sqrt{x} \) is always non-negative, and it is multiplied by \( -1 \) in this function, the maximum value occurs at the endpoint and is \( -4 \).

As \( x \) increases, \( \sqrt{x + 3} \) increases, but because of the negative sign in front of the square root, the values decrease further below \( -4 \).

Thus, the range is:
\[
(-\infty, -4]
\]

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### Summary:
1. **The endpoint of the graph is at** \( (-3, -4) \).
2. **The domain of the function is** \( [-3, \infty) \).
3. **The range of the function is** \( (-\infty, -4] \).

Would you like more details or have any questions? Here are five related questions to explore:

1. How would the graph change if the function were \( f(x) = \sqrt{x + 3} - 4 \)?
2. What would the range be if the constant "-4" were changed to "+4"?
3. Can you determine the horizontal shift of the graph based on \( x + 3 \)?
4. How would you describe the overall shape of square root functions in general?
5. What are the transformations applied to the parent function \( \sqrt{x} \) in this case?

**Tip:** Always consider the transformations of basic functions (like square roots) when analyzing graphs to make sense of their shifts, stretches, and reflections.

The function f(x) = -√(x + 3) - 4 is graphed below. Find the endpoint, domain, and range of the function.

Learn how to find the endpoint, domain, and range of the square root function f(x) = -√(x + 3) - 4. This example walks through the solution, highlighting key concepts in graphing transformations and square root functions. Suitable for students in Grades 9-11.

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