To analyze the transformations of the square root function $f(x) = -2\sqrt{x} - 5$, we start with the parent function $g(x) = \sqrt{x}$ and identify how $f(x)$ differs.

Steps to Identify Transformations:

1. Reflection:

   • The negative sign ($-$) before $2$ causes a vertical reflection across the $x$-axis.

2. Vertical Stretch:

   • The coefficient $2$ scales the function vertically by a factor of $2$, making the graph steeper.

3. Vertical Translation:

   • The $-5$ outside the square root shifts the graph down by 5 units.

Combined Effect:

• Starting from $g(x) = \sqrt{x}$:
  1. Reflect it vertically about the $x$-axis.
  2. Stretch it vertically by a factor of 2.
  3. Shift it 5 units downward.

Description of the Graph:

The graph of $f(x) = -2\sqrt{x} - 5$:

• Begins at the same $x$-intercept as $g(x)$ (at $x = 0$).
• Decreases (instead of increasing) because of the reflection.
• Is steeper due to the vertical stretch.
• Lies 5 units below the original reflected and stretched function.

Let me know if you’d like to see a visual sketch or additional details! 😊

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Expanding Questions:

1. What happens if the $-2$ is replaced with $+2$?
2. How would the transformations change if the equation included a horizontal shift, like $f(x) = -2\sqrt{x-3} - 5$?
3. What are the domain and range of $f(x) = -2\sqrt{x} - 5$?
4. How does the vertical stretch affect the rate of change of the function?
5. How would the graph of $g(x) = \sqrt{x}$ change if reflected across the $y$-axis instead?

Tip: When analyzing transformations, always separate horizontal and vertical changes to simplify the process!