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.

Would you like me to generate this graph for you?

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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 $x$.