Use this guide to understand how parameters a, h, and k affect the graphs of parent functions.

Let's solve the problem step by step.

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Problem:

Graph the function:
$$
f(x) = -2\sqrt{x + 1} - 3
$$

We are also asked to identify the transformations and summarize the rules for graphing this function.

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Step 1: Identify the Parent Function

The parent function here is:
$$
f(x) = \sqrt{x}
$$
This is a square root function, which has a domain of $ x \geq 0 $, and its graph starts at the origin (0, 0) and increases slowly to the right.

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Step 2: Rewrite the Function in Standard Form

Compare:
$$
f(x) = -2\sqrt{x + 1} - 3
$$
with the general form of a transformed square root function:
$$
y = a\sqrt{x - h} + k
$$

So, let’s match the terms:

- $ a = -2 $
- $ h = -1 $ (because $ x + 1 = x - (-1) $)
- $ k = -3 $

So:
- $ a = -2 $: vertical stretch by factor 2 and reflection over the x-axis
- $ h = -1 $: horizontal shift left by 1 unit
- $ k = -3 $: vertical shift down by 3 units

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Step 3: Determine Domain and Range

#### Domain:
The expression under the square root must be non-negative:
$$
x + 1 \geq 0 \Rightarrow x \geq -1
$$
So, domain: $ [-1, \infty) $

#### Range:
The basic $ \sqrt{x} $ has range $ [0, \infty) $

Apply transformations:
- Multiply by $-2$: reflects and stretches → output becomes $ (-\infty, 0] $
- Then subtract 3: shifts down → final range: $ (-\infty, -3] $

So, range: $ (-\infty, -3] $

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Step 4: Create a Table of Values

We'll pick values of $ x $ starting from $ x = -1 $ (the left endpoint), then increase.

| $ x $ | $ x + 1 $ | $ \sqrt{x + 1} $ | $ -2\sqrt{x + 1} $ | $ f(x) = -2\sqrt{x + 1} - 3 $ |
|--------|------------|---------------------|-----------------------|-------------------------------|
| -1 | 0 | 0 | 0 | -3 |
| 0 | 1 | 1 | -2 | -5 |
| 3 | 4 | 2 | -4 | -7 |
| 8 | 9 | 3 | -6 | -9 |

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Step 5: Graph the Function

Plot the points from the table:
- $ (-1, -3) $: starting point
- $ (0, -5) $
- $ (3, -7) $
- $ (8, -9) $

The graph will start at $ (-1, -3) $ and go downward to the right, with a shape similar to a square root but reflected and stretched.

It should look like the left half of a parabola flipped upside down and shifted.

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Step 6: Summary of Transformations

| Transformation | Effect |
|----------------|--------|
| $ \sqrt{x + 1} $ | Shift left 1 unit |
| $ -2\sqrt{x + 1} $ | Reflect over x-axis, vertical stretch by factor 2 |
| $ -2\sqrt{x + 1} - 3 $ | Shift down 3 units |

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Final Answer:

#### Graphing Steps:
1. Start with parent function $ y = \sqrt{x} $
2. Shift left 1 unit → $ y = \sqrt{x + 1} $
3. Reflect over x-axis and stretch vertically by 2 → $ y = -2\sqrt{x + 1} $
4. Shift down 3 units → $ y = -2\sqrt{x + 1} - 3 $

#### Key Features:
- Domain: $ [-1, \infty) $
- Range: $ (-\infty, -3] $
- Vertex (starting point): $ (-1, -3) $
- Direction: Decreasing from left to right

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✔ Final Graph Description (since no image can be drawn here):

On the coordinate plane:
- Plot the point $ (-1, -3) $
- Use the table values to plot more points: $ (0, -5), (3, -7), (8, -9) $
- Draw a smooth curve starting at $ (-1, -3) $ and decreasing to the right, approaching negative infinity as $ x \to \infty $

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Summary Box (as requested):

> SUMMARY:
>
> The function $ f(x) = -2\sqrt{x + 1} - 3 $ is a transformation of the square root function.
>
> - Parent function: $ \sqrt{x} $
> - Transformations:
> - Horizontal shift: left 1 unit (due to $ x + 1 $)
> - Vertical stretch by factor 2
> - Reflection over the x-axis (due to negative sign)
> - Vertical shift: down 3 units
> - Domain: $ x \geq -1 $ → $ [-1, \infty) $
> - Range: $ y \leq -3 $ → $ (-\infty, -3] $
> - Graph: Starts at $ (-1, -3) $, decreases slowly to the right.

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✔ You can now sketch the graph using the table and transformations above.