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Chapter 1: Problem 87

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

Short Answer

Expert verified
The graph of the function \(f(x)=x^{2}-2|x|-3\) looks more like a W. It comes from combining a parabolic curve and a V-shaped curve in a particular way.

Step by step solution

01

Understand the equation

The given equation, \(f(x)=x^{2}-2|x|-3\), has two main parts: quadratic part (\(x^{2}\)) and absolute value part (\(-2|x|\)). It's important to understand that absolute value makes all its input positive.
02

Graph the absolute value term

First, graph the absolute value term on its own. The graph of absolute value function \(y=|x|\) is a V-shaped curve which is always nonnegative. Multiplying it by -2, the graph of \(y=-2|x|\) is still a V-shaped curve but it opens downwards.
03

Graph the quadratic term

Then, graph the quadratic part (\(y=x^{2}\)). Quadratic functions have a U-shaped curve, called a parabola, which opens upwards or downwards based on the coefficient of \(x^{2}\) term.
04

Combine the absolute value and quadratic graphs

Subtract the absolute value graph, \(y=-2|x|\), from the quadratic graph, \(y=x^{2}\), which means at each point \(x\), the y-coordinate of the resultant graph would be the y-coordinate of the quadratic graph minus the y-coordinate of the absolute value graph. Doing so would produce the graph of the function \(y=x^{2}-2\|x\|\).
05

Graph the final equation

Now subtract 3 from the y-coordinate of each point of the resultant graph from the previous step. Finally you would get the graph of the function \(f(x)=x^{2}-2|x|-3\).

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Most popular questions from this chapter

Each function has two or more independent yariables. Given \(g(x, y)=2 x^{2}-|y|+3,\) find a. \(g(3,-4)\) b. \(g(-1,2)\) c. \(g(0,-5)\) d. \(g\left(\frac{1}{2},-\frac{1}{4}\right)\) e. \(g(c, 3 c), c>0\) f. \(g(c+5, c-2), c<0\)

The perimeter of a rectangle is 27 centimeters, and its area is 35 square centimeters. Find the length and the width of the rectangle.

The notation \(\left.f(x)\right|_{a} ^{b}\) is used to denote the difference \(f(b)-f(a) .\) That is, $$\left.f(x)\right|_{a} ^{b}=f(b)-f(a)$$ Evaluate \(\left.f(x)\right|_{0} ^{b}\) for the given function \(f\) and the indicated values of \(a\) and \(b\). $$f(x)=-3 x+2 ;\left.f(x)\right|_{4} ^{7}$$

The length of the side of a square has been measured accurately to within 0.01 foot. This measured length is 4.25 feet. a. Write an absolute value inequality that describes the relationship between the actual length of each side of the square s and its measured length. b. Solve the absolute value inequality you found in part a. for s.

Use interval notation to express the solution set of each inequality. $$|3-2 x| \leq 5$$
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