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Chapter 8: Problem 112

Use a graphing utility to graph \(y=\sqrt{x}, y=\sqrt{x}+4\) and \(y=\sqrt{x}-3\) in the same \([-1,10,1]\) by \([-10,10,1]\) viewing rectangle. Describe one similarity and one difference that you observe among the graphs.

Short Answer

Expert verified
The similarity among the graphs of \(y = \sqrt{x}\), \(y = \sqrt{x} + 4\), and \(y = \sqrt{x} - 3\) is that they all have the characteristic shape of a square root function. However, the difference lies in their vertical positions. The graph of \(y = \sqrt{x} + 4\) is shifted 4 units upwards, and the graph of \(y = \sqrt{x} - 3\) is shifted 3 units downwards compared to the original \(y = \sqrt{x}\) function.

Step by step solution

01

Graphing the basic equation

Start by graphing the first equation, \(y = \sqrt{x}\). Note that the graph of a square root function typically starts at the origin (0,0) and curves upward to the right.
02

Graphing the transformed functions

Next, graph the other two equations: \(y = \sqrt{x} + 4\) and \(y = \sqrt{x} - 3\). These are transformations of the original function. Adding or subtracting a constant from the basic square root function essentially slides or shifts the graph up or down. So, \(y = \sqrt{x} + 4\) should be identical to \(y = \sqrt{x}\), but shifted upward by 4 units. Similarly, \(y = \sqrt{x} - 3\) would be shifted downward by 3 units.
03

Identifying similarities and differences

Once you have all three graphs, observe them within the given viewing rectangle. By comparison, notice the similarities and differences among the graphs. The similarity is that all three equations have the same general shape, which is characteristic of the graph of a square root function. The difference is that the location or position of each graph along the y-axis varies depending on the added or subtracted constant, resulting in the entire graph sliding or shifting up or down.

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