Graph the following function by moving the green and blue dots (if necessary).
y, equals, minus, 2, square root of, x, end square root
y=−2
x
​

The given function is \( y = -2\sqrt{x} \).

To graph this function, let's break it down step-by-step:

1. **Basic Form**: The base function is \( y = \sqrt{x} \). This is a square root function, and it only exists for \( x \geq 0 \). The graph of \( y = \sqrt{x} \) starts at the origin (0,0) and increases slowly, curving upwards.

2. **Transformation**:
   - The multiplication by \(-2\) causes two transformations:
     - The negative sign flips the graph over the x-axis (reflection).
     - The multiplication by 2 stretches the graph vertically by a factor of 2, making the curve steeper.

### Key Points on the Graph:
- **At \( x = 0 \)**: \( y = -2\sqrt{0} = 0 \). So, the graph passes through the origin (0,0).
- **At \( x = 1 \)**: \( y = -2\sqrt{1} = -2 \). The graph passes through the point (1, -2).
- **At \( x = 4 \)**: \( y = -2\sqrt{4} = -4 \). The graph passes through the point (4, -4).

### Characteristics of the Graph:
- The graph exists only for \( x \geq 0 \).
- It curves downward due to the negative sign, and the curve becomes steeper because of the multiplication by 2.

### Graph Description:
- The graph starts at the origin (0, 0), and as \( x \) increases, \( y \) becomes more negative.
  
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**Further Questions:**

1. How does the graph of \( y = \sqrt{x} \) change with different coefficients?
2. What is the domain and range of \( y = -2\sqrt{x} \)?
3. How does the reflection across the x-axis affect other functions?
4. Can we apply similar transformations to other types of functions?
5. What happens to the graph when you change the coefficient of \( \sqrt{x} \) to a positive value?

**Tip**: Always remember that the square root function only has real outputs for non-negative values of \( x \).

Graph the following function by moving the green and blue dots (if necessary). y = -2√x

Explore how to graph the square root function y = -2√x by reflecting it across the x-axis and applying a vertical stretch. This detailed step-by-step guide walks through key transformations and graphing techniques. Suitable for students in Grades 9-11.

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