Graph each function.
$\text{ƒ}\left(x\right)=-\text{√}x-2$

Verified step by step guidance

1. Identify the base function to graph, which is the square root function \(f(x) = \sqrt{x}\). This function is defined for \(x \geq 0\) and its graph starts at the origin \((0,0)\), increasing slowly to the right.
2. Apply the negative sign in front of the square root, changing the function to \(f(x) = -\sqrt{x}\). This reflects the graph of \(\sqrt{x}\) across the x-axis, so the graph will start at \((0,0)\) and decrease as \(x\) increases.
3. Apply the vertical shift by subtracting 2, resulting in \(f(x) = -\sqrt{x} - 2\). This moves the entire graph down by 2 units, so the starting point moves from \((0,0)\) to \((0,-2)\).
4. Determine the domain and range of the function. The domain remains \(x \geq 0\) because the square root is only defined for non-negative \(x\). The range shifts to \(y \leq -2\) because the graph is reflected and shifted down.
5. Plot key points such as \((0,-2)\), \((1,-3)\), and \((4,-4)\) by substituting values of \(x\) into the function, then sketch the curve starting at \((0,-2)\) and decreasing gently to the right.

Key Concepts

Square Root Function

The square root function, denoted as √x, outputs the non-negative value whose square is x. Its domain is all non-negative real numbers (x ≥ 0), and its graph starts at the origin (0,0) and increases slowly to the right. Understanding this base function is essential before applying transformations.

Vertical Reflection and Translation

Multiplying the square root function by -1 reflects its graph across the x-axis, flipping it upside down. The '-2' outside the square root shifts the entire graph downward by 2 units. Recognizing these transformations helps in accurately sketching the modified function.

Domain and Range of Transformed Functions

The domain of ƒ(x) = -√x - 2 remains x ≥ 0 since the square root is undefined for negative inputs. The range changes due to reflection and translation; here, it becomes y ≤ -2 because the graph is flipped and shifted down. Identifying domain and range ensures the graph is correctly bounded.