In Exercises 64–66, begin by graphing the square root function, f(x) = √x. Then use transformations of this graph to graph the given function. g(x) = √(x + 3)

Verified step by step guidance

1. Start by understanding the parent 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. Identify the transformation in the given function \(g(x) = \sqrt{x + 3}\). Notice that inside the square root, \(x\) is replaced by \(x + 3\), which indicates a horizontal shift.
3. Recall that replacing \(x\) by \(x + c\) inside the function shifts the graph horizontally to the left by \(c\) units. Here, \(c = 3\), so the graph of \(f(x)\) shifts 3 units to the left.
4. To graph \(g(x)\), take the graph of \(f(x) = \sqrt{x}\) and move every point 3 units left. For example, the starting point of \(f(x)\) at \((0,0)\) moves to \((-3,0)\) for \(g(x)\).
5. Finally, sketch the transformed graph starting at \((-3,0)\) and increasing to the right, maintaining the same shape as the original square root function.

Key Concepts

Square Root Function

The square root function, f(x) = √x, outputs the non-negative root of x and is defined for x ≥ 0. Its graph starts at the origin (0,0) and increases slowly, forming a curve in the first quadrant. Understanding this base graph is essential before applying transformations.

Function Transformations

Function transformations involve shifting, stretching, compressing, or reflecting the graph of a base function. For g(x) = √(x + 3), the '+3' inside the square root causes a horizontal shift of the graph to the left by 3 units. Recognizing how changes inside the function affect the graph is key.

Horizontal Shifts

A horizontal shift moves the graph left or right along the x-axis. For a function f(x + c), the graph shifts left by c units if c > 0, and right if c < 0. In g(x) = √(x + 3), the '+3' shifts the graph of √x three units left, changing the domain accordingly.