Problem 75 - 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. $$g(x)=\frac{1}{2} \sqrt{x+2}$$

Step-by-Step Solution

Answer

The function \(g(x)=\frac{1}{2}\sqrt{x+2}\) can be graphed by first graphing the base square root function \(f(x)=\sqrt{x}\), then shifting everything 2 units to the left and stretching vertically by a factor of 1/2.

Step 1: Graph the base function

First graph the square root function \(f(x)=\sqrt{x}\). The graph starts at the point (0,0) and then grows gradually. As this is a square root function, its domain is all non-negative numbers and its range is also all non-negative numbers.

Step 2: Apply the horizontal shift

The presence of \(x+2\) inside the square root function indicates a horizontal shift of the base function by 2 units to the left. So, everything on the graph should be moved 2 units to the left.

Step 3: Apply the vertical stretch

The factor of 1/2 in front of the square root function is a vertical stretch by a factor of 1/2. Every y-coordinate should be multiplied by 1/2. As a result, for every x in the graph, the value of y is reduced by half. It grows more slowly than the base square root function and reaches any particular Y value at a point further to the right.

Step 4: Graph the final function

After applying the horizontal shift and vertical stretch to the base function, you would get the graph of \(g(x)=\frac{1}{2}\sqrt{x+2}\). It starts from the point (-2,0) and expands more slowly than the base function. This is due to the vertical stretch of 1/2 making the graph half as tall at each point.