Problem 76 - 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. $$g(x)=2 \sqrt{x+1}$$

Step-by-Step Solution

Answer

The graph of the function \(g(x)=2 \sqrt{x+1}\) is obtained by taking the square root function, shifting it 1 unit to the left, and vertically dilating by a factor of 2.

Step 1: Graph the Basic Function \(f(x)=\sqrt{x}\)

First, graph the basic square root function, \(f(x)=\sqrt{x}\). The graph starts at the origin (0, 0) and increases slowly towards positive infinity.

Step 2: Apply Horizontal Translation

Next, observe the \(x+1\) part in the given function \(g(x)=2 \sqrt{x+1}\). This represents a horizontal shift or translation of the graph. Since it is \(x+1\), it means the graph \(f(x)=\sqrt{x}\) needs to be shifted 1 unit to the left. The function becomes \(f(x)=\sqrt{x+1}\).

Step 3: Apply Vertical Dilation

The last transformation comes from the coefficient '2' in the given function \(g(x)=2 \sqrt{x+1}\). This '2' scales the output of the square root function by a factor of 2. This is a vertical dilation of the graph. The final graph of the function is then \(g(x)=2 \sqrt{x+1}\). This graph starts 1 unit to the left of the origin (at (-1,0)) and increases at twice the rate of the basic square root function.