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Chapter 3: Problem 8

Exer. 7-8: Describe the set of all points \(P(x, y)\) in a coordinate plane that satisfy the given condition. (a) \(y=-2\) (b) \(x=-4\) (c) \(x / y<0\) (d) \(x y=0\) (e) \(y>1\) (f) \(y=0\)

Short Answer

Expert verified
(a) Horizontal line y=-2; (b) Vertical line x=-4; (c) Points in 2nd & 4th quadrants; (d) Coordinates on x-axis & y-axis; (e) Area above y=1; (f) x-axis points.

Step by step solution

01

Step 1

Identify the condition in part (a), which is \( y = -2 \), and understand that this represents a horizontal line in the coordinate plane. Thus, the set of points \( P(x, y) \) is described as all points where the y-coordinate is always \(-2\), regardless of the x-coordinate.
02

Step 2

For part (b), \( x = -4 \) represents a vertical line. The set of points \( P(x, y) \) consists of all points where the x-coordinate is \(-4\), with any y-coordinate.
03

Step 3

In part (c), the condition \( \frac{x}{y} < 0 \) implies that x and y have opposite signs. This means: if \( x > 0 \) then \( y < 0 \) and if \( x < 0 \) then \( y > 0 \). Therefore, the set of points includes all points in the second and fourth quadrants of the coordinate plane.
04

Step 4

Part (d) says \( x y = 0 \). For this to be true, either \( x = 0 \) or \( y = 0 \). The set of points consists of the entire x-axis (\( y = 0 \)) and y-axis (\( x = 0 \)), including the origin.
05

Step 5

For part (e), \( y > 1 \) represents the region above the horizontal line \( y = 1 \). The set of points is all points where the y-coordinate is greater than 1, for any x-coordinate.
06

Step 6

In part (f), \( y = 0 \) describes the x-axis. This means that the set of points includes all points where the y-coordinate is zero, accepting any x-coordinate.

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Key Concepts

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

Coordinate Plane
The coordinate plane is a two-dimensional surface where we can graphically represent points, lines, and shapes. It is defined by two perpendicular axes: the horizontal axis, commonly referred to as the x-axis, and the vertical axis, known as the y-axis. Together, they create a framework in which any point can be described using a pair of coordinates, \(x, y\). Each point on this plane corresponds to a unique pair consisting of an x-coordinate and a y-coordinate.
  • The x-coordinate tells us how far a point is from the vertical y-axis, to the left or to the right.
  • The y-coordinate informs us of the point's distance from the horizontal x-axis, either up or down.
Understanding the coordinate plane is fundamental when dealing with geometric figures and equations, as it allows us to visualize and solve mathematical problems with ease.
Horizontal Line
A horizontal line on a coordinate plane is a straight line that runs from left to right and remains constant at a particular y-coordinate. This mean that no matter how far you extend the line, each point on it will have the same y-value.
  • In the equation form, a horizontal line is expressed as \(y = c\), where \(c\) is a constant.
  • This line is parallel to the x-axis and never intersects it.
For instance, if we consider the equation \(y = -2\), it means the horizontal line will pass through any point where the y-coordinate is \(-2\), such as points \((-3, -2)\), \(0, -2)\), and even \(5, -2)\). Such lines are crucial for defining limits and boundaries within the coordinate system.
Vertical Line
A vertical line in a coordinate plane runs up and down, maintaining a constant x-coordinate along its course. This type of line signifies that at every point through which it passes, the x-value does not change, although the y-values can vary.
  • Mathematically, a vertical line is represented by \(x = c\), where \(c\) is the constant x-coordinate of every point on the line.
  • It runs parallel to the y-axis and never crosses it.
For example, the equation \(x = -4\) describes a vertical line where every point has an x-coordinate of \(-4\). This includes points like \((-4, 0)\) and \((-4, 3)\). Understanding vertical lines is essential, especially when determining how two points relate spatially on the plane.
Quadrants
The coordinate plane is divided into four sections known as quadrants, each determined by the positive and negative values of the x and y coordinates. These quadrants help categorize the points based on their relative positions to the origin \(0,0\):
  • First Quadrant: Both x and y coordinates are positive (e.g., \(2, 3\)).
  • Second Quadrant: x is negative while y is positive (e.g., \(-2, 3\)).
  • Third Quadrant: Both x and y coordinates are negative (e.g., \(-2, -3\)).
  • Fourth Quadrant: x is positive while y is negative (e.g., \(2, -3\)).
Knowing which quadrant a point resides in is crucial for solving many geometric problems. It also helps in understanding relationships between variables in given contexts, such as inequality conditions. For example, when given the condition \(\frac{x}{y}<0\), points are found in the second and fourth quadrants because x and y have opposite signs in these areas.

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Most popular questions from this chapter

A spherical balloon is being inflated at a rate of \(\frac{9}{2} \pi \mathrm{ft}^{3} / \mathrm{min}\). Express its radius \(r\) as a function of time \(t\) (in minutes), assuming that \(r=0\) when \(t=0\).

The diagonal \(d\) of a cube is the distance between two opposite vertices. Express \(d\) as a function of the edge \(x\) of the cube. (Hint: First express the diagonal \(y\) of a face as a function of \(x\).)

Exer. 13-26: Sketch, on the same coordinate plane, the graphs of \(f\) for the given values of \(c\). (Make use of symmetry, shifting, stretching, compressing, or reflecting.) $$ f(x)=(x+c)^{3} ; \quad c=-2,1,2 $$

Exer. 21-34: Find (a) \((f \circ g)(x)\) and the domain of \(f \circ g\) and (b) \((g \circ f)(x)\) and the domain of \(g \circ f\). $$ f(x)=-x^{2}+1, \quad g(x)=\sqrt{x} $$

Exer. 21-34: Find (a) \((f \circ g)(x)\) and the domain of \(f \circ g\) and (b) \((g \circ f)(x)\) and the domain of \(g \circ f\). $$ f(x)=x^{2}-4, \quad g(x)=\sqrt{3 x} $$
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