use translations to graph the given function.                                                                                                                                L (x) = √x-5

To graph the function \( L(x) = \sqrt{x - 5} \) using translations, we need to understand how the basic function \( f(x) = \sqrt{x} \) is transformed.

### Step-by-Step Process:

1. **Start with the basic function**: The basic function is \( f(x) = \sqrt{x} \), which starts at the origin \((0, 0)\) and gradually increases as \( x \) increases, forming a curve that opens to the right.

2. **Identify the transformation**: The given function is \( L(x) = \sqrt{x - 5} \). This is a horizontal translation of the basic function \( f(x) = \sqrt{x} \).

3. **Horizontal Shift**: The term \((x - 5)\) inside the square root function indicates a horizontal shift to the right by 5 units. This means every point on the graph of \( f(x) = \sqrt{x} \) is moved 5 units to the right.

### Graphing the Function:

- **Domain**: The function \( \sqrt{x - 5} \) is defined only when the expression inside the square root is non-negative, i.e., \( x - 5 \geq 0 \). Thus, the domain is \( x \geq 5 \).
  
- **Range**: Since \( \sqrt{x - 5} \) outputs non-negative values, the range is \( y \geq 0 \).

- **Key Points**:
  - Start at \( (5, 0) \) because when \( x = 5 \), \( L(x) = 0 \).
  - As \( x \) increases, the function value increases as the square root grows.

Here is a quick sketch of the graph of \( L(x) = \sqrt{x - 5} \):

1. Plot the point \( (5, 0) \).
2. For \( x = 6 \), \( L(6) = \sqrt{6 - 5} = 1 \). So plot the point \( (6, 1) \).
3. For \( x = 9 \), \( L(9) = \sqrt{9 - 5} = 2 \). So plot the point \( (9, 2) \).
4. Draw a smooth curve through these points, starting from \( (5, 0) \) and gradually increasing to the right.

Would you like a visual representation of this graph, or is there anything specific you'd like me to expand upon?

### Related Questions:
1. How would the graph change if the function were \( L(x) = \sqrt{x + 5} \)?
2. What is the effect of vertical translations on the function \( f(x) = \sqrt{x} \)?
3. How would you find the inverse of the function \( L(x) = \sqrt{x - 5} \)?
4. How does adding a coefficient in front of the square root, such as \( 2\sqrt{x - 5} \), affect the graph?
5. What are the effects of reflections on the graph of square root functions?

**Tip:** When graphing functions using transformations, always start by identifying the basic function and then apply transformations like shifts, reflections, or stretches step-by-step.

use translations to graph the given function. L(x) = √x-5

Learn how to graph the square root function L(x) = √(x - 5) using horizontal translations. This step-by-step guide explains transformations, the domain and range, and key points for plotting the graph. Suitable for students in Grades 9-11.

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