Problem 67 - 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. $$g(x)=\sqrt{x}+2$$

Step-by-Step Solution

Answer

The graph of \(g(x)=\sqrt{x} + 2\) is the same as the graph of \(f(x)=\sqrt{x}\), but shifted 2 units upwards.

Step 1: Graph the basic square root function

Begin by graphing the square root function \(f(x)=\sqrt{x}\). The graph starts from the origin (0,0) and extends towards positive x, looking like the half part of a parabola lying on its side.

Step 2: Understand the transformation

Next, analyze the transformation required to obtain the function \(g(x)\) from \(f(x)\). Here, \(g(x)= f(x) + 2\), which means that every point on the graph of \(f(x)\) is shifted up by 2 units to result in the graph of \(g(x)\). This transformation is a simple vertical shift, and it does not change the shape of the graph.

Step 3: Apply the transformation to the graph

Finally, apply the vertical shift to the graph of the square root function. This means shifting every point of the original graph upwards by 2 units. No points are left out, and because the shape of the graph does not change, it still looks like a half parabola, just translated upwards.