Problem 78 - 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. $$h(x)=\sqrt{x+1}-1$$

Step-by-Step Solution

Answer

The graph of \(h(x)=\sqrt{x+1}-1\) can be obtained by taking the graph of the parent function \(f(x)=\sqrt{x}\), shifting it 1 unit to the left and then 1 unit downwards.

Step 1: Understand the initial square root function

The function \(f(x) = \sqrt{x}\) starts from the origin (0,0) and as 'x' increases, 'y' also increases slowly. As 'x' gets larger, the curve of the function becomes flatter.

Step 2: Recognize and apply horizontal transformation

The \(+1\) inside the square root function represents a horizontal shift. This is a shift to the left by 1 unit. For every 'x' in the given function \(h(x)=\sqrt{x+1}-1\), let it be \(x+1\), then graph the points as they would be in the parent function \(f(x)=\sqrt{x}\), but 1 unit to the left.

Step 3: Recognize and apply vertical transformation

The \(-1\) outside the square root function represents a vertical shift. This is a downward shift by 1 unit. Following the horizontal transformation, shift each point on the graph vertically downward by 1 unit to match the given function \(h(x)=\sqrt{x+1}-1\).

Step 4: Draw the graph of the function

After applying all transformations, connect the transformed points with a smooth curve to graph the function \(h(x)=\sqrt{x+1}-1\). Care should be taken to maintain the basic shape of the start of the square root function.