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Chapter 2: Problem 78

Begin by graphing the square root function, \(f(x)=\sqrt{x} .\) Then use transformations of this graph to graph the given function. $$h(x)=\sqrt{x+1}-1$$

Short Answer

Expert verified
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 by step solution

01

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.
02

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.
03

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 \(h(x)=\sqrt{x+1}-1\).
04

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.

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