Question 1175341
Let's break down the transformations step-by-step to graph h(x) = 1 - f(2(4-x)) from f(x) = √x.

**1. Function f(x) = √x**

* This is the basic square root function.

**2. Inside Transformations (2(4-x))**

* **-x:** Reflection across the y-axis. This gives us √(-x).
* **4-x:** Horizontal shift 4 units to the right. This gives us √(-(x-4)) = √(4-x).
* **2(4-x):** Horizontal compression by a factor of 1/2. This gives us √(2(4-x)) = √(8-2x).

**3. Outside Transformations (1 - ...)**

* **f(2(4-x)):** we have now √(8-2x)
* **-f(2(4-x)):** Reflection across the x-axis. This gives us -√(8-2x).
* **1 - f(2(4-x)):** Vertical shift 1 unit upward. This gives us 1 - √(8-2x).

**Therefore, h(x) = 1 - √(8 - 2x).**

**Graphing the Transformations**

1.  **Start with f(x) = √x.**
2.  **Reflect across the y-axis (√(-x)).**
3.  **Shift 4 units right (√(4-x)).**
4.  **Compress horizontally by 1/2 (√(8-2x)).**
5.  **Reflect across the x-axis (-√(8-2x)).**
6.  **Shift 1 unit up (1 - √(8-2x)).**

**Finding Key Points**

* **Domain:** 8 - 2x ≥ 0 => 8 ≥ 2x => x ≤ 4. So the domain is (-∞, 4].
* **x-intercept:** 1 - √(8 - 2x) = 0 => 1 = √(8 - 2x) => 1 = 8 - 2x => 2x = 7 => x = 3.5.
* **y-intercept:** h(0) = 1 - √(8 - 2(0)) = 1 - √8 ≈ 1 - 2.828 ≈ -1.828.
* **Endpoint:** At x = 4, h(4) = 1 - √(8 - 2(4)) = 1 - √0 = 1.

**Graph Image (using Gyazo)**

[Gyazo Link: [https://i.gyazo.com/14299775f560e90c67e812d614838495.png](https://www.google.com/search?q=https://i.gyazo.com/14299775f560e90c67e812d614838495.png) ]

**Explanation of the Graph:**

* The graph starts at the point (4, 1).
* It decreases as x decreases, moving to the left.
* It crosses the x-axis at x = 3.5.
* It crosses the y-axis at approximately y=-1.828.