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Chapter 8: Problem 9

Sketch the graph of the inequality. $$y x^{2} \geq 1$$

Short Answer

Expert verified
Shade the region above the curve of \( y = \frac{1}{x^2} \).

Step by step solution

01

Rewrite the Inequality

First, rewrite the inequality to make it easier for analysis. The given inequality is \( y x^2 \geq 1 \). We can write it in terms of \( y \): \[ y \geq \frac{1}{x^2} \] This gives us a function \( y = \frac{1}{x^2} \), which represents the boundary of the inequality.
02

Analyze the Boundary Function

Now, analyze the function \( y = \frac{1}{x^2} \). This is a rational function that has the following characteristics: 1. The function is undefined at \( x = 0 \) due to division by zero.2. For \( x > 0 \) and \( x < 0 \), the function is defined and positive.3. The function approaches infinity as \( x \) approaches zero from either direction (left or right). 4. The function approaches zero as \( x \) moves away from the origin towards positive or negative infinity.
03

Plot the Boundary Function

Next, sketch the graph of the boundary function \( y = \frac{1}{x^2} \). It will be a curve in the first and second quadrants where:- As \( x \to 0^+ \) or \( x \to 0^- \), \( y \to \infty \).- As \( x \to \infty \) or \( x \to -\infty \), \( y \to 0 \).
04

Shade the Appropriate Region

Since the inequality is \( y \geq \frac{1}{x^2} \), shade the region above the curve \( y = \frac{1}{x^2} \). This includes all points where the value of \( y \) is equal to or greater than \( \frac{1}{x^2} \). This region will include lines going outward from both branches of the hyperbola.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Rational Functions
A rational function is a type of function that is represented by the ratio of two polynomials. It is expressed in the form \( f(x) = \frac{P(x)}{Q(x)} \), where both \( P(x) \) and \( Q(x) \) are polynomials, and \( Q(x) eq 0 \). In the context of the exercise, the rational function is given by \( y = \frac{1}{x^2} \). Here, \( P(x) = 1 \) and \( Q(x) = x^2 \). Some important properties of rational functions include:
  • They can have vertical asymptotes where the denominator is zero, as seen when \( x = 0 \) since \( \frac{1}{x^2} \) is undefined at this point.
  • They can have horizontal or slant asymptotes, which help in describing the end behavior of the function.
  • They often result in curves or graphs with distinct characteristics, such as hyperbolas.
The function \( y = \frac{1}{x^2} \) is a rational function that's undefined at \( x = 0 \), and exhibits behavior typical of hyperbolas, which we'll discuss more later.
Quadrants
In the Cartesian plane, a quadrant is one of the four sections created by dividing the plane with an x-axis and a y-axis. These axes intersect at the origin \( (0, 0) \). Starting from the upper right and moving counter-clockwise, the quadrants are numbered:
  • First Quadrant (I): Where x and y are both positive.
  • Second Quadrant (II): Where x is negative and y is positive.
  • Third Quadrant (III): Where both x and y are negative.
  • Fourth Quadrant (IV): Where x is positive and y is negative.
For the function \( y = \frac{1}{x^2} \), the relevant quadrants are the first and second quadrants. This is because as \( x \) approaches zero (from either the positive or negative side), \( y \) increases toward infinity, forming a curve existing solely in these two quadrants. Thus, when graphing rational functions like hyperbolas, understanding quadrants helps delineate where parts of the function will reside.
Shading Regions
When graphing inequalities such as \( y \geq \frac{1}{x^2} \), a crucial step is shading the appropriate region of the graph. This process indicates the set of points that satisfy the inequality conditions. In this inequality:- We graph the boundary curve first, \( y = \frac{1}{x^2} \).- Then we determine which side of this curve satisfies \( y \geq \frac{1}{x^2} \), which is the region above the graph.To ensure you have shaded the correct region:
  • Choose a test point located in each possible shading region (e.g., \( (1, 2) \)).
  • Substitute the coordinates of these points into the inequality.
  • If the inequality holds true for a test point, shade the region containing this point.
This method visually expands the set of solutions, highlighting not only the boundary curve, but also the infinite number of points above it that satisfy the inequality.
Hyperbola
A hyperbola is a type of smooth curve lying on a plane, defined by its distinct geometry. It's usually the result of slicing a double conic section in a unique way. In terms of functions and graphs, a rational function like \( y = \frac{1}{x^2} \) exhibits hyperbolic characteristics, as parts of its graph resemble hyperbolas. Characteristics of hyperbolas include:
  • They consist of two disconnected, mirror-image branches, as seen in this exercise.
  • Each branch approaches, but never intersects, specific asymptotes, which are lines the curve gets arbitrarily close to.
  • The center at \( x = 0 \) represents a point of infinite discomfort, a point the branches veer away from.
The given rational function illustrates these characteristics by forming branches in the first and second quadrants. These branches approach asymptotes as \( x \) moves toward zero (vertically) and move towards zero (horizontally) as \( x \) increases positively or negatively, embodying the hyperbola's nature.

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Most popular questions from this chapter

Use matrices to solve the system. $$\left\\{\begin{aligned} x+3 y-z &=0 \\ -x-2 y+z &=0 \\ -2 x+y+3 z &=0 \end{aligned}\right.$$

Without expanding, explain why the statement is true. $$\left|\begin{array}{lll} 2 & 3 & 1 \\ 0 & 4 & 5 \\ 1 & 1 & 4 \end{array}\right|=-\left|\begin{array}{lll} 1 & 1 & 4 \\ 0 & 4 & 5 \\ 2 & 3 & 1 \end{array}\right|$$

Filling a pool A swimming pool can be filled by three pipes, \(A, B,\) and \(C .\) Pipe \(A\) alone can fill the pool in 8 hours. If pipes A and C are used together, the pool can be filled in 6 hours; if \(\mathrm{B}\) and \(\mathrm{C}\) are used together, it takes 10 hours. How long does it take to fill the pool if all three pipes are used?

Let \(I=I_{3}\) and let \(f(x)=|A-x I| .\) Find (a) the polynomial \(f(x)\) and (b) the zeros of \(f(x).\) \(A=\left[\begin{array}{rrr}3 & 2 & 2 \\ 1 & 0 & 2 \\ -1 & -1 & 0\end{array}\right]\)

Without expanding, explain why the statement is true. $$\left|\begin{array}{rrr} 1 & -1 & 2 \\ 1 & 2 & -1 \\ 1 & -1 & 2 \end{array}\right|=0$$
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