The equation you've provided is \( y = -\sqrt{x} + 4 \). Let's break down this equation and analyze it.

### Understanding the Equation

1. **Square Root Function**: The expression \(\sqrt{x}\) represents the square root of \(x\). The square root function is defined for all \(x \geq 0\), meaning we cannot take the square root of a negative number (if we are considering real numbers only).

2. **Negative Sign in Front**: The negative sign before the square root, \(-\sqrt{x}\), indicates that the square root values will be inverted. Normally, the square root function \(\sqrt{x}\) is increasing, but \(-\sqrt{x}\) will be decreasing. This creates a reflection over the \(x\)-axis of the standard square root function.

3. **Vertical Shift**: The "+4" at the end of the expression represents a vertical shift upwards by 4 units. So, every point on the graph of \(-\sqrt{x}\) is shifted up by 4 units.

### Graph Analysis

- **Domain**: The domain of \(y = -\sqrt{x} + 4\) is all \(x \geq 0\) because the square root function is only defined for non-negative \(x\). 

- **Range**: Since \(\sqrt{x}\) outputs non-negative numbers (0 and positive values), \(-\sqrt{x}\) will output 0 or negative values. After shifting up by 4, the largest value of \(y\) is 4 when \(x = 0\), and as \(x\) increases, \(y\) decreases. Therefore, the range of \(y = -\sqrt{x} + 4\) is \(y \leq 4\).

### Key Points on the Graph

1. **At \(x = 0\)**: 
   \[
   y = -\sqrt{0} + 4 = 4
   \]
   The point \((0,

Explore the equation y = -√(x) + 4 in detail with this comprehensive analysis. Learn about the square root function, graphing functions, and vertical shifts. Suitable for advanced high school students and anyone interested in mathematical functions and graphs.

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