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Chapter 3: Problem 20

Plot the following points in a rectangular coordinate system. For each point, name the quadrant in which it lies or the axis on which it lies. $$\left(4,-\frac{10}{3}\right)$$

Short Answer

Expert verified
The point \(4, -\frac{10}{3}\) lies in Quadrant IV.

Step by step solution

01

Identify the coordinates

The given point is \((4, -\frac{10}{3})\). The first number in the ordered pair is the x-coordinate (4), and the second number is the y-coordinate \(-\frac{10}{3}\).
02

Determine the signs of the coordinates

The x-coordinate is positive (4), and the y-coordinate is negative \(-\frac{10}{3}\).
03

Locate the point in the coordinate system

In a rectangular (Cartesian) coordinate system, a point \(x, y\) where x is positive and y is negative is located in Quadrant IV.
04

Plot the point

On the graph, find the x-value at 4 on the x-axis and drop down to the y-coordinate \(-\frac{10}{3}\) to plot the point \(4, -\frac{10}{3}\).
05

State the quadrant

Since both x is positive and y is negative, the point lies in Quadrant IV.

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Key Concepts

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

coordinate system
The coordinate system, also known as the Cartesian coordinate system, is a framework used for plotting points in a plane using coordinates. Each point is defined by an ordered pair \(x, y\). The system consists of two perpendicular lines called the x-axis (horizontal) and the y-axis (vertical). These axes intersect at a point called the origin, represented by \(0, 0\). To plot a point, you move right or left along the x-axis and then up or down along the y-axis.
quadrants
In the Cartesian coordinate system, the plane is divided into four sections, known as quadrants. Each quadrant is identified by the signs of the x and y coordinates of the points within it. Starting from the top-right and moving counterclockwise:
  • Quadrant I (top-right): Both x and y are positive.
  • Quadrant II (top-left): x is negative, y is positive.
  • Quadrant III (bottom-left): Both x and y are negative.
  • Quadrant IV (bottom-right): x is positive, y is negative.
The point \(4, -\frac{10}{3}\) is in Quadrant IV, as the x-coordinate is positive and the y-coordinate is negative.
plotting points
Plotting points involves locating points on the Cartesian plane based on their coordinates \(x, y\). Here鈥檚 how to do it:
  1. Identify the x-coordinate (first number) and y-coordinate (second number).
  2. Locate the x-value on the x-axis.
  3. From the x-value, move vertically to the y-value on the y-axis.
  4. Mark the point where the x and y values intersect.
In the example \(4, -\frac{10}{3}\), find '4' on the x-axis. Then, move down because \(-\frac{10}{3}\) is negative. Mark the point where these meet.
x and y coordinates
Coordinates are numerical values that determine the location of a point in the Cartesian plane. The first number in an ordered pair \((x, y)\) is the x-coordinate, which shows how far to move left or right from the origin. The second number is the y-coordinate, indicating how far to move up or down. For \(4, -\frac{10}{3}\), the number '4' instructs to move right from the origin because it is positive, and \(-\frac{10}{3}\) means to move down since it is negative.

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Most popular questions from this chapter

Determine which of the ordered pairs \((1,3),(-2,5)\) \((-6,-4),\) and \((7,-8)\) satisfy each compound or absolute value inequality. $$y>-x+1 \text { or } y<4 x$$

A gravel dealer charges \(\$ 50\) plus \(\$ 30\) per cubic yard for delivering a truckload of gravel. Express the total cost \(C(n)\) in dollars as a function of the number of cubic yards delivered \(n .\) Find \(C(12)\).

Find the equation of line l in each case and then write it in standard form with integral coefficients. Line \(l\) goes through \((-3,-1)\) and is parallel to \(y=2 x+6\).

Graph \(y=-1 x, y=-2 x, y=-3 x,\) and \(y=-4 x\) together in the standard viewing window. These equations are all of the form \(y=m x\). What effect does decreasing \(m\) have on the graph of the equation? What are the slopes of these four lines?

Find an equation of the line that goes through the given point and has the given slope. Give the answer in slope-intercept form. See Example 5 (-5, 150) with slope -30
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