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: $(-\infty, -4]$

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.