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Chapter 7: Problem 13

Plot the given point in a rectangular coordinate system. \(\left(-3,-1 \frac{1}{2}\right)\)

Short Answer

Expert verified
The point \(-3, -1 \frac{1}{2}\) on a rectangular coordinate system would appear 3 units to the left of the origin on the x-axis and \(\frac{3}{2}\) units below the origin on the y-axis.

Step by step solution

01

Identify the coordinates

Identify the x and y coordinates from the given point \(-3, -1 \frac{1}{2}\). Here, the x-coordinate, which signifies horizontal movement, is -3, and the y-coordinate, which signifies vertical movement, is \(-1 \frac{1}{2}\). The negative x-coordinate means moving 3 units to the left from the origin (0,0) on the horizontal axis while the negative y-coordinate means moving \(\frac{3}{2}\) units downwards from the origin on the vertical axis.
02

Plot the X-coordinate

The x-coordinate is -3. Starting from the origin (0,0), move 3 units to the left on the x-axis since the x-coordinate is negative.
03

Plot the Y-coordinate

The y-coordinate is \(-1 \frac{1}{2}\). From the point on the x-axis (-3,0), move \(\frac{3}{2}\) units downward on the y-axis because the y-coordinate is negative.
04

Mark the final point

Mark the final point at (-3,-1.5) representing the original given coordinates. This is the point of intersection after moving 3 units to the left from the origin on the x-axis and \(\frac{3}{2}\) units downward on the y-axis.

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Key Concepts

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

Plotting Points
To plot points on a rectangular coordinate system, it's essential to comprehend both coordinates, which dictate the position of the point in a two-dimensional space. A point's position is expressed as an ordered pair
  • (x, y), where the first value is the x-coordinate,
  • the second is the y-coordinate.
In this context, the rectangular coordinate system is like a map that helps identify where a point is located based on how far it is from the origin. The origin is the central point
  • commonly denoted as (0, 0),
  • marking where the x-axis and y-axis intersect.
Each coordinate tells us how far the point is from these intersecting axes. To accurately plot a point, start at the origin, then move according to the coordinates.
X-Coordinate
The x-coordinate directs how far to move horizontally from the origin to plot a point. This value tells us whether to move left or right on the x-axis.
If the x-coordinate is negative, like -3, you move to the left.
If it is positive, you move to the right.
  • Start at the origin (0, 0).
  • Move horizontally along the x-axis.
  • For an x-coordinate of -3, move three units left.
This horizontal movement places you directly on the x-axis at the new location in relation to its numerical value. This step sets the stage for placing a point in relation to the y-coordinate.
Y-Coordinate
Understanding the y-coordinate is crucial for determining vertical movement after positioning according to the x-coordinate. The y-coordinate indicates how far up or down to move.
  • From your position on the x-axis,
  • move depending on the y-coordinate's sign and value,
  • since a negative y-coordinate like y = -1 \(\frac{1}{2}\)means moving downward.
In this case, starting from the location of (-3,0), descend 1.5 units vertically. This downward movement positions you exactly where the final point of (-3,-1 \(\frac{1}{2}\)) should be plotted on the graph.
Graphing Skills
Developing graphing skills involves repeated practice and familiarizing yourself with moving along the axes based on given coordinates. Here are some tips:
  • Always start at the origin when plotting new points.
  • Visualize the movement required by each coordinate.
  • For negative values, remember it indicates moving in the opposite direction (left for x, down for y).
  • Use a ruler to ensure movements are precise and neat on paper.
Continual practice plotting various points enhances understanding of geometric relationships and spatial awareness. Mastering these skills allows for accurate graphing and a strong foundation in mathematical concepts.

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Most popular questions from this chapter

A television manufacturer makes rear-projection and plasma televisions. The profit per unit is \(\$ 125\) for the rearprojection televisions and \(\$ 200\) for the plasma televisions. a. Let \(x=\) the number of rear-projection televisions manufactured in a month and \(y=\) the number of plasma televisions manufactured in a month. Write the objective function that describes the total monthly profit. b. The manufacturer is bound by the following constraints: \- Equipment in the factory allows for making at most 450 rear-projection televisions in one month. \- Equipment in the factory allows for making at most 200 plasma televisions in one month. \- The cost to the manufacturer per unit is \(\$ 600\) for the rear-projection televisions and \(\$ 900\) for the plasma televisions. Total monthly costs cannot exceed \(\$ 360,000 .\) Write a system of three inequalities that describes these constraints. c. Graph the system of inequalities in part (b). Use only the first quadrant and its boundary, because \(x\) and \(y\) must both be nonnegative. d. Evaluate the objective function for total monthly profit at each of the five vertices of the graphed region. [The vertices should occur at \((0,0),(0,200),(300,200)\), \((450,100)\), and \((450,0)\).] e. Complete the missing portions of this statement: The television manufacturer will make the greatest profit by manufacturing rear-projection televisions each month and plasma televisions each month. The maximum monthly profit is $\$$

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The graphs of solution sets of systems of inequalities involve finding the intersection of the solution sets of two or more inequalities. By contrast, in Exercises 43-44, you will be graphing the union of the solution sets of two inequalities. Graph the union of \(x-y \geq-1\) and \(5 x-2 y \leq 10\).

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Many elevators have a capacity of 2000 pounds. a. If a child averages 50 pounds and an adult 150 pounds, write an inequality that describes when \(x\) children and \(y\) adults will cause the elevator to be overloaded. 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?
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