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Chapter 7: Problem 1

In Exercises 1-20, plot the given point in a rectangular coordinate system. \((1,4)\)

Short Answer

Expert verified
The point \((1,4)\) is plotted on the Cartesian coordinate system by starting at the origin, moving 1 unit right along the x-axis, and then 4 units upwards along the y-axis. The final position is our point \((1,4)\).

Step by step solution

01

Understand the Coordinates

The provided point is an ordered pair in the form \((x,y)\) given as \((1,4)\), where '1' is the x-coordinate and '4' is the y-coordinate. The x-coordinate represents the horizontal distance from the origin while the y-coordinate represents the vertical distance from the origin.
02

Plotting the X-Coordinate

Start at the origin (0,0), move 1 unit to the right along the x-axis. This is because the x-coordinate is 1, and it's positive so we move to the right.
03

Plotting the Y-Coordinate

From the position after step 2, move 4 units upwards along the y-axis. This is because the y-coordinate is 4, and it's positive so we move up.
04

Mark The Point

Mark the point and label it as \((1,4)\). The marked point represents the location of \((1,4)\) on the Cartesian coordinate system.

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Key Concepts

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

Plotting Points
Plotting points in a rectangular coordinate system is essential to graphing in mathematics. A point is defined using an ordered pair of numbers, such as (1,4). These numbers provide the exact location of the point in the system.

To plot points:
  • Start at the origin, which is the intersection of the x-axis and the y-axis, often denoted as (0,0).
  • Use the x-coordinate to determine how far to move horizontally. A positive x-value signifies moving right, while a negative value indicates moving left.
  • Next, use the y-coordinate to move vertically. A positive y-value means moving upwards and a negative value means moving downwards.
  • Finally, place a mark at the intersection of these movements and label the point accordingly, for instance, (1,4).
Plotting points accurately helps visualize equations and understand spatial relationships within the coordinate system.
Ordered Pairs
In a coordinate system, every point is represented by an ordered pair, usually in the form (x, y). The essence of ordered pairs is the specific sequence they follow, where the x-value always comes before the y-value. This sequence is crucial since it specifies the point's precise location on the graph.

An ordered pair denotes a relationship:
  • The x-coordinate tells you the position along the horizontal axis (how far from the center you are moving left or right).
  • The y-coordinate indicates the position along the vertical axis (how far up or down you are going from the horizontal line).
With our example, the ordered pair (1,4) identifies exactly one unique spot on the graph, granting a clear way of distinguishing different points. This ordering ensures there is no confusion about the point's actual place in the coordinate plane, which would happen if they were switched around.
Cartesian Coordinate System
The Cartesian coordinate system is a fundamental element in plotting points and graphing equations. It consists of two perpendicular lines, forming a plane divided into four quadrants.

Essential features of the Cartesian coordinate system:
  • The horizontal axis is the x-axis, where positive values stretch to the right and negative values to the left.
  • The vertical axis is the y-axis, with positive values extending upwards and negatives heading downwards.
  • Where these axes cross each other at (0,0) is called the origin.
This system allows us to represent visual and numerical relationships clearly. By employing ordered pairs and plotting points, we can create graphs of lines, curves, and various shapes, making the abstract concepts of algebra and geometry tangible and understandable. This system is named after the French mathematician Ren茅 Descartes, who developed this form of graphing.

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Most popular questions from this chapter

If \(x\) represents height, in inches, and y represents weight, in pounds, the healthy weight region can be modeled by the following system of linear inequalities: $$ \left\\{\begin{array}{l} 5.3 x-y \geq 180 \\ 4.1 x-y \leq 140 \end{array}\right. $$ Use this information to solve Exercises 45-48. Is a person in this age group who is 6 feet tall weighing 205 pounds within the healthy weight region?

a. Create a scatter plot for the data in each table. b. Use the shape of the scatter plot to determine if the data are best modeled by a linear function, an exponential function, a logarithmic function, or a quadratic function. $$ \begin{array}{|c|r|} \hline \boldsymbol{x} & \boldsymbol{y} \\ \hline 0 & 5 \\ \hline 1 & 3 \\ \hline 2 & 1 \\ \hline 3 & -1 \\ \hline 4 & -3 \\ \hline \end{array} $$

A patient is not allowed to have more than 330 milligrams of cholesterol per day from a diet of eggs and meat. Each egg provides 165 milligrams of cholesterol. Each ounce of meat provides 110 milligrams. a. Write an inequality that describes the patient's dietary restrictions for \(x\) eggs and \(y\) ounces of meat. b. Graph the inequality. Because \(x\) and \(y\) must be positive, limit the graph to quadrant I only. c. Select an ordered pair satisfying the inequality. What are its coordinates and what do they represent in this situation?

Use the directions for Exercises 9-14 to graph each quadratic function. Use the quadratic formula to find \(x\)-intercepts, rounded to the nearest tenth. \(f(x)=-3 x^{2}+6 x-2\)

Graph the solution set of each system of inequalities. \(\left\\{\begin{array}{l}x-y \leq 1 \\ x \geq 2\end{array}\right.\)
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