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Chapter 1: Problem 90

Use a graphing utility. Graph: \(f(x)=\left|x^{2}-2 x\right|-3\)

Short Answer

Expert verified
The graph is a 'w' shaped curve with vertices at points (0,-3) and (2,-3).

Step by step solution

01

Break Down the Absolute Function

To graph this function, first need to break down the absolute value into two separate equations. \(f(x)=x^{2}-2 x-3\) when \(x^{2}-2x 鈮� 0\) and \(f(x) = - (x^{2}-2x) - 3\) when \(x^{2}-2x < 0\).
02

Identify the points where function changes

Find out where \(x^{2}-2x = 0\) this will give the vertices of the absolute value function, which in turn indicates where the function changes from the positive part of the quadratic function to the negative part and vice versa. To find these, solve the quadratic equation \(x^{2}-2x = 0\), which gives \(x = 0, 2\).
03

Graph the Two Parts of the Function

Now graph the two separate functions \(f(x) = x^{2}-2x - 3\) and \(f(x) = - (x^{2}-2x) - 3\) Keeping in mind that each function only applies to either the left side or the right side of the vertices points found in step 2. As an example, for \(x < 0\) and \(x > 2\), use \(f(x) = x^{2}-2x - 3\) and for \(0 鈮� x 鈮� 2\), use \(f(x) = - (x^{2}-2x) - 3\). You may use any graphical calculator or software to draw these two graphs.
04

Combine the Two Parts

Finally, merge the two parts of the graph to get the graph of the given function \(f(x)=\left|x^{2}-2 x\right|-3\). The graph is the upper part of the U-shaped graph from the first half and the upside-down U shape from the second half. This gives the entire shape of the graph for the absolute value function.

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