In Exercises 67-80, begin by graphing the square root function, f(x) = √x. Then use transformations of this graph to graph the given function. h(x)=-√(x + 1)

Verified step by step guidance

1. Start by graphing the basic square root function $f(x)=\sqrt{x}$. This graph is a curve that starts at the origin (0,0) and increases slowly to the right.
2. Next, consider the transformation $\sqrt{x+1}$. This represents a horizontal shift of the graph of $\sqrt{x}$ to the left by 1 unit. So, the starting point of the graph moves from (0,0) to (-1,0).
3. Now, apply the negative sign in front of the square root: $-\sqrt{x+1}$. This reflects the graph of $\sqrt{x+1}$ over the x-axis, flipping it upside down.
4. The resulting graph of $h(x)=-\sqrt{x+1}$ starts at (-1,0) and decreases as it moves to the right, mirroring the shape of the original square root function but in the opposite direction.
5. Finally, ensure the domain of $h(x)$ is considered. Since you cannot take the square root of a negative number, the domain of $h(x)$ is $x\ge -1$.

Key Concepts

Square Root Function

The square root function, f(x) = √x, is defined for x ≥ 0 and produces non-negative outputs. Its graph is a curve that starts at the origin (0,0) and increases gradually, reflecting the relationship between x and its square root. Understanding this function is crucial as it serves as the base for applying transformations.

Graph Transformations

Graph transformations involve shifting, reflecting, stretching, or compressing the graph of a function. In this case, the function h(x) = -√(x + 1) involves a horizontal shift to the left by 1 unit and a reflection across the x-axis. Mastery of these transformations allows for the accurate graphing of new functions based on known ones.

Reflection Across the X-Axis

Reflection across the x-axis occurs when the output values of a function are negated. For the function h(x) = -√(x + 1), this means that every point on the graph of f(x) = √(x + 1) is mirrored over the x-axis. This transformation changes the direction of the graph, flipping it upside down, which is essential for accurately representing the function.