document.write( "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. \r
\n" ); document.write( "\n" ); document.write( "Use Gyazo (or another tool) to capture your graph as an image. Insert your image or the Gyazo link here. \n" ); document.write( "
Algebra.Com's Answer #850619 by CPhill(1959)\"\" \"About 
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Let's break down the transformations step-by-step to graph h(x) = 1 - f(2(4-x)) from f(x) = √x.\r
\n" ); document.write( "\n" ); document.write( "**1. Function f(x) = √x**\r
\n" ); document.write( "\n" ); document.write( "* This is the basic square root function.\r
\n" ); document.write( "\n" ); document.write( "**2. Inside Transformations (2(4-x))**\r
\n" ); document.write( "\n" ); document.write( "* **-x:** Reflection across the y-axis. This gives us √(-x).
\n" ); document.write( "* **4-x:** Horizontal shift 4 units to the right. This gives us √(-(x-4)) = √(4-x).
\n" ); document.write( "* **2(4-x):** Horizontal compression by a factor of 1/2. This gives us √(2(4-x)) = √(8-2x).\r
\n" ); document.write( "\n" ); document.write( "**3. Outside Transformations (1 - ...)**\r
\n" ); document.write( "\n" ); document.write( "* **f(2(4-x)):** we have now √(8-2x)
\n" ); document.write( "* **-f(2(4-x)):** Reflection across the x-axis. This gives us -√(8-2x).
\n" ); document.write( "* **1 - f(2(4-x)):** Vertical shift 1 unit upward. This gives us 1 - √(8-2x).\r
\n" ); document.write( "\n" ); document.write( "**Therefore, h(x) = 1 - √(8 - 2x).**\r
\n" ); document.write( "\n" ); document.write( "**Graphing the Transformations**\r
\n" ); document.write( "\n" ); document.write( "1. **Start with f(x) = √x.**
\n" ); document.write( "2. **Reflect across the y-axis (√(-x)).**
\n" ); document.write( "3. **Shift 4 units right (√(4-x)).**
\n" ); document.write( "4. **Compress horizontally by 1/2 (√(8-2x)).**
\n" ); document.write( "5. **Reflect across the x-axis (-√(8-2x)).**
\n" ); document.write( "6. **Shift 1 unit up (1 - √(8-2x)).**\r
\n" ); document.write( "\n" ); document.write( "**Finding Key Points**\r
\n" ); document.write( "\n" ); document.write( "* **Domain:** 8 - 2x ≥ 0 => 8 ≥ 2x => x ≤ 4. So the domain is (-∞, 4].
\n" ); document.write( "* **x-intercept:** 1 - √(8 - 2x) = 0 => 1 = √(8 - 2x) => 1 = 8 - 2x => 2x = 7 => x = 3.5.
\n" ); document.write( "* **y-intercept:** h(0) = 1 - √(8 - 2(0)) = 1 - √8 ≈ 1 - 2.828 ≈ -1.828.
\n" ); document.write( "* **Endpoint:** At x = 4, h(4) = 1 - √(8 - 2(4)) = 1 - √0 = 1.\r
\n" ); document.write( "\n" ); document.write( "**Graph Image (using Gyazo)**\r
\n" ); document.write( "\n" ); document.write( "[Gyazo Link: [https://i.gyazo.com/14299775f560e90c67e812d614838495.png](https://www.google.com/search?q=https://i.gyazo.com/14299775f560e90c67e812d614838495.png) ]\r
\n" ); document.write( "\n" ); document.write( "**Explanation of the Graph:**\r
\n" ); document.write( "\n" ); document.write( "* The graph starts at the point (4, 1).
\n" ); document.write( "* It decreases as x decreases, moving to the left.
\n" ); document.write( "* It crosses the x-axis at x = 3.5.
\n" ); document.write( "* It crosses the y-axis at approximately y=-1.828.
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