Problem 79 - 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. $$g(x)=2 \sqrt{x+2}-2$$

Step-by-Step Solution

Answer

The graph of the function \(g(x)=2\sqrt{x+2}-2\) will look like the graph of the basic square root function \(f(x)=\sqrt{x}\), but it will be stretched vertically by a factor of 2, shifted 2 units to the left, and then shifted 2 units down.

Step 1: Graph basic function

Start by graphing the basic square root function \(f(x)=\sqrt{x}\). This graph starts at the origin (0, 0) and increases slowly, curving upwards to the right.

Step 2: Identify transformations

Next, identify the transformations present in the function \(g(x)=2\sqrt{x+2}-2\). From this, recognize that there are three transformations: a vertical stretch by a factor of 2, a horizontal shift 2 units to the left, and a vertical shift 2 units down.

Step 3: Apply transformations

Apply the transformations one at a time to the graph of the basic function in the following order: horizontal shift, vertical stretch, and vertical shift. A horizontal shift of 2 units to the left means moving every point on \(f(x)=\sqrt{x}\) two places to the left. A vertical stretch by a factor of 2 involves multiplying the y-coordinates of every point on the graph by 2. Lastly, vertical shift 2 units down means moving every point on the transformed graph two places down.

Step 4: Join the transformed points

After transforming all the points, join these points to create a continuous graph. This final graph represents the function \(g(x)=2\sqrt{x+2}-2\).