graph the equation shown below by transforming the given graph of the parent function

To graph the equation \( y = 2\sqrt{x} \) by transforming the graph of the parent function \( y = \sqrt{x} \), follow these steps:

### 1. **Identify the Parent Function:**
   - The parent function is \( y = \sqrt{x} \).
   - The graph of \( y = \sqrt{x} \) starts at the origin \((0,0)\) and curves upward to the right.

### 2. **Understand the Transformation:**
   - The equation \( y = 2\sqrt{x} \) introduces a vertical stretch by a factor of 2.
   - This means that for every point \((x, y)\) on the graph of the parent function \( y = \sqrt{x} \), the corresponding point on the graph of \( y = 2\sqrt{x} \) will be \((x, 2y)\).

### 3. **Apply the Transformation:**
   - Start with key points on the graph of \( y = \sqrt{x} \):
     - \((0, 0)\)
     - \((1, 1)\)
     - \((4, 2)\)
     - \((9, 3)\)
   - Apply the vertical stretch by multiplying the \( y \)-coordinate of each point by 2:
     - \((0, 0) \rightarrow (0, 0)\)
     - \((1, 1) \rightarrow (1, 2)\)
     - \((4, 2) \rightarrow (4, 4)\)
     - \((9, 3) \rightarrow (9, 6)\)

### 4. **Plot the Transformed Points:**
   - Plot the points \((0, 0)\), \((1, 2)\), \((4, 4)\), and \((9, 6)\) on a coordinate plane.
   - Draw a smooth curve through these points to complete the graph.

### 5. **Sketch the Graph:**
   - The graph should start at the origin \((0, 0)\) and curve upwards to the right, steeper than the parent function due to the vertical stretch.

### Visualization:
Imagine a graph where the curve from \( y = \sqrt{x} \) is stretched vertically. The curve for \( y = 2\sqrt{x} \) will look similar but rise faster.

Would you like to see a graph generated from this transformation?

### Related Questions:
1. How does the graph of \( y = 3\sqrt{x} \) compare to \( y = 2\sqrt{x} \)?
2. What happens to the graph of \( y = \sqrt{x} \) if you apply a horizontal shift?
3. Can you describe the graph of \( y = \sqrt{-x} \)?
4. What is the effect of a negative sign on the graph of \( y = -2\sqrt{x} \)?
5. How does adding a constant, like \( y = 2\sqrt{x} + 1 \), affect the graph?

### Tip:
When transforming graphs, remember that multiplying by a constant affects the steepness, while adding or subtracting constants shifts the graph.

Learn how to graph the equation y = 2sqrt(x) by transforming the parent function y = sqrt(x). This detailed solution explains the vertical stretch transformation applied to the parent function's graph, including plotting key points and sketching the resulting curve. Suitable for high school students.

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