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Chapter 3: Problem 78

Give an example of an ordered pair whose location is in (or on) a. quadrant I b. quadrant II c. quadrant III d. quadrant IV e. \(x\) -axis f. \(y\) -axis

Short Answer

Expert verified
Quadrant I: (2, 3); Quadrant II: (-4, 5); Quadrant III: (-3, -1); Quadrant IV: (6, -2); x-axis: (5, 0); y-axis: (0, 7).

Step by step solution

01

Identify Quadrant I

A point in Quadrant I has both positive x and y coordinates. For example, the ordered pair \((2, 3)\) lies in Quadrant I.
02

Identify Quadrant II

A point in Quadrant II has a negative x coordinate and a positive y coordinate. For example, the ordered pair \((-4, 5)\) is located in Quadrant II.
03

Identify Quadrant III

A point in Quadrant III has both negative x and y coordinates. For example, the ordered pair \((-3, -1)\) lies in Quadrant III.
04

Identify Quadrant IV

A point in Quadrant IV has a positive x coordinate and a negative y coordinate. An example of such a pair is \((6, -2)\).
05

Locate on x-axis

A point on the x-axis has a zero y coordinate. For instance, \((5, 0)\) is located on the x-axis.
06

Locate on y-axis

A point on the y-axis has a zero x coordinate. An example is \((0, 7)\), which lies on the y-axis.

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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 locate points using ordered pairs. These pairs consist of two numbers written in the form \(x, y\). This grid-like system helps us to easily identify the exact position of any point in space by specifying its horizontal and vertical distance from the origin.
  • The horizontal axis is called the x-axis.
  • The vertical axis is known as the y-axis.
The point where these axes intersect is called the origin, represented by the ordered pair \(0, 0\). This setup allows us to navigate and plot points with precision, making the coordinate plane a fundamental tool in mathematics.
Quadrants
The coordinate plane is divided into four sections, known as quadrants. Each quadrant is labeled with Roman numerals and has distinct characteristics based on the signage of the x and y values. Here's a breakdown:
  • Quadrant I: Both x and y coordinates are positive.
  • Quadrant II: The x coordinate is negative, while the y coordinate is positive.
  • Quadrant III: Both x and y coordinates are negative.
  • Quadrant IV: The x coordinate is positive, and the y coordinate is negative.
This division into quadrants helps in systematically analyzing and categorizing points on the coordinate plane. For example, the point \(2, 3\) lies in Quadrant I, indicating its position falls where both coordinates are positive.
X-Axis and Y-Axis
In the coordinate system, the x-axis and y-axis are the two main reference lines used to plot points. The x-axis runs horizontally, while the y-axis runs vertically, intersecting at the origin \(0, 0\). Together, they form the framework of the coordinate plane.
  • The x-axis is the boundary between Quadrants I & IV on the top and II & III below.
  • The y-axis is the boundary between Quadrants I & II on the right and III & IV on the left.
Points that lie directly on the x-axis will have a y-coordinate of zero, such as \(5, 0\). Conversely, points on the y-axis have an x-coordinate of zero, like \(0, 7\). This makes it easy to identify which axis a point lies on simply by looking at these zeroes.
Positive and Negative Coordinates
Knowing whether a coordinate is positive or negative helps in determining its location on the coordinate plane. Coordinates are labeled as positive or negative based on their relationship to the origin.
  • Positive x-coordinates are to the right of the y-axis.
  • Negative x-coordinates are to the left of the y-axis.
  • Positive y-coordinates are above the x-axis.
  • Negative y-coordinates are below the x-axis.
Understanding this concept of positive and negative coordinates allows us to predict where a point will lie on the plane. For instance, the ordered pair \((-4, 5)\) resides in Quadrant II due to the negative x and positive y values.

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Most popular questions from this chapter

Assume each exercise describes a linear relationship. Write the equations in slope-intercept form. In \(2004,\) there were approximately 83,000 gaselectric hybrid vehicles sold in the United States. In \(2007,\) there were approximately 353,000 such vehicles sold. (Source: Energy Information Administration, Department of Energy) a. Assume the relationship between years past 2004 and the number of vehicles sold is linear over this period. Write an equation describing the relationship between time and the number of gas-electric hybrid vehicles sold. Use ordered pairs of the form (years past \(2004,\) number of vehicles sold). b. Use this equation to estimate the number of gas-electric hybrid sales in \(2009 .\)

Write an equation in standard form of the line that contains the point (-1,2) and is parallel to (has the same slope as) the line \(y=3 x-1\)

Solve. See Example 4. The table shows the amount of money (in billions of dollars) that Americans spent on their pets for the years shown. (Source: American Pet Products Manufacturers Association) $$ \begin{array}{|c|c|} \hline \text { Year } & \begin{array}{c} \text { Pet-Related Expenditures } \\ \text { (in billions of dollars) } \end{array} \\ \hline 2005 & 36.3 \\ \hline 2006 & 38.5 \\ \hline 2007 & 41.2 \\ \hline 2008 & 43.4 \\ \hline \end{array} $$ a. Write this paired data as a set of ordered pairs of the form (year, pet- related expenditures). b. In your own words, write the meaning of the ordered pair (2007,41.2) c. Create a scatter diagram of the paired data. \(\mathrm{Be}\) sure to label the axes appropriately. d. What trend in the paired data does the scatter diagram show?

Graph each inequality. $$ y<\frac{2}{5} x-3 $$

Determine whether (1,1) is included in each graph. $$ x>3 $$
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