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Question 1175341: Consider f(x)= √x and h(x)= 1-f(2(4-x)). Apply the transformations to f(x) to graph the function h(x). Show your work.
Use Gyazo (or another tool) to capture your graph as an image. Insert your image or the Gyazo link here.
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! 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.
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