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. r(x) = 2√(x + 2)

Verified step by step guidance

1. Start with the basic square root function \(f(x) = \sqrt{x}\), which is graphed starting at the point \((0,0)\) and increases slowly as \(x\) increases.
2. Identify the transformation inside the square root: \(r(x) = 2\sqrt{x + 2}\). The expression \(x + 2\) inside the square root indicates a horizontal shift to the left by 2 units.
3. Apply the horizontal shift by moving every point on the graph of \(f(x) = \sqrt{x}\) two units to the left. For example, the point \((0,0)\) on \(f(x)\) moves to \((-2,0)\) on \(r(x)\).
4. Next, observe the coefficient 2 outside the square root, which means a vertical stretch by a factor of 2. Multiply all the \(y\)-values of the shifted graph by 2.
5. Combine these transformations: first shift the graph of \(f(x)\) left by 2 units, then stretch it vertically by a factor of 2 to get the graph of \(r(x) = 2\sqrt{x + 2}\).

Key Concepts

Square Root Function

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

Horizontal Shifts of Functions

A horizontal shift occurs when a constant is added or subtracted inside the function's argument, such as in f(x + c). For f(x + 2), the graph shifts 2 units to the left. This transformation changes the starting point of the graph along the x-axis without altering its shape.

Vertical Stretching of Functions

Multiplying a function by a constant greater than 1, like 2√(x + 2), stretches the graph vertically. This means all y-values are multiplied by 2, making the graph steeper. Vertical stretching affects the output values but does not shift the graph horizontally.