Problem 77 - College Algebra Essentials

Question

Begin by graphing the square root function, \(f(x)=\sqrt{x} .\) Then use transformations of this graph to graph the given function. $$h(x)=\sqrt{x+2}-2$$

Step-by-Step Solution

Answer

The function \(h(x) = \sqrt{x+2} - 2\) is a transformation of the square root function \(f(x) = \sqrt{x}\). The transformation involves a horizontal shift to the left by 2 units and a vertical shift downwards by 2 units. The graph starts from the point (-2,-2).

Step 1: Original Function

Begin with the original function \(f(x) = \sqrt{x}\). The graph of this function starts from the point (0,0) and rises slowly, moving to the right along the x-axis.

Step 2: Horizontal translation

Observe the given function \(h(x) = \sqrt{x+2} - 2\). The first transformation is \(x + 2\). This shifts every point in the function \(f(x)\) to the left by 2 units. The graph now starts at the point (-2,0).

Step 3: Vertical translation

The second transformation from the function \(h(x) = \sqrt{x+2} - 2\) is \(-2\). This will shift every point down by 2 units. This means that the graph which previously started at (-2,0) will now start at (-2,-2). Apply this transformation to all points on the graph.

Step 4: New Function Graph

The final graph has been transformed from the original function \(f(x) = \sqrt{x}\) by moving 2 units to the left and 2 units downwards. This will be the graph of function \(h(x) = \sqrt{x+2} - 2\)