Problem 69 - 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 identical to the graph of \(f(x)=\sqrt{x}\), but shifted 2 units to the left. So it starts at (-2,0) and then continues to rise as x increases, in the same way that the original graph does, but from an earlier starting point.

Step 1: Graph the original square root function

Draw the graph of \(f(x)=\sqrt{x}\). This function starts at point (0,0) and continues to rise as x increases, creating a curve that rises slower as x becomes larger.

Step 2: Understand the transformation

The transformation from \(f(x)=\sqrt{x}\) to \(g(x)=\sqrt{x+2}\) involves a shift to the left by 2 units. This is because the \(x+2\) inside the square root in \(g(x)\) means that the function will now return a square root value for \(x\) values that are 2 less than those of the original function \(f(x)\). So, for example, where \(f(4)=2\), now \(g(2)=2\). The effect of this transformation is to shift the entire graph of \(f(x)\) 2 units to the left.

Step 3: Graph the transformed function

Draw the graph of \(g(x)=\sqrt{x+2}\). It will look similar to the graph of \(f(x)=\sqrt{x}\), but shifted 2 units to the left. So the new graph will begin at point (-2,0) and continue upwards, mimicking the rise of the original function as x increases, but starting 2 units earlier.