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Chapter 5: Problem 4

For each of the following, graph the given points. $$ (-4,-2) $$

Short Answer

Expert verified
Graph the point \((-4, -2)\) by moving 4 units left and 2 units down from the origin.

Step by step solution

01

- Understanding Cartesian Coordinates

In Cartesian coordinates, each point is defined by an ordered pair \((x, y)\). The first number (\(x\)) represents the horizontal position, and the second number (\(y\)) represents the vertical position.
02

- Identify the Coordinates

The given point is \((-4, -2)\). Here, the x-coordinate is \(-4\) and the y-coordinate is \(-2\).
03

- Plot the Point on the Graph

To plot the point \((-4, -2)\):1. Start at the origin \((0,0)\).2. Move 4 units to the left on the x-axis (because the x-coordinate is \(-4\)).3. Move 2 units down on the y-axis (because the y-coordinate is \(-2\)).
04

- Draw and Label the Point

Mark the point at the position reached in step 3 and label it as \((-4, -2)\) on the graph.

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Key Concepts

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

Cartesian coordinates
The Cartesian coordinate system is a foundational concept in mathematics, used to represent points in a plane using two numbers. These numbers, called coordinates, are written as an ordered pair \((x, y)\). The first number, \(x\), indicates the position along the horizontal axis, and the second number, \(y\), indicates the position along the vertical axis.

  • The \((x, y)\) format represents each point precisely.
  • The Cartesian plane is divided into four quadrants.
  • Each point has a unique location identified by its coordinates.

This system was invented by the French mathematician Ren茅 Descartes, and it allows us to visualize mathematical concepts and perform geometric operations easily. Understanding this system is crucial for graphing functions, solving algebraic equations, and many other mathematical tasks.
x-axis and y-axis
The x-axis and y-axis are the two perpendicular lines that create the Cartesian coordinate system. The x-axis runs horizontally, while the y-axis runs vertically.

  • The intersection point of the x-axis and y-axis is called the origin, represented as \( (0,0) \).
  • The x-axis is used to measure horizontal distance.
  • The y-axis is used to measure vertical distance.

When plotting a point like \((-4, -2)\), the x-coordinate \(-4\) tells you to move 4 units to the left from the origin, and the y-coordinate \(-2\) tells you to move 2 units down. By understanding how the x-axis and y-axis work, you can accurately locate any point on the Cartesian plane.
Plotting points
Plotting points on a Cartesian plane involves a few simple steps:

  • Start at the origin \((0,0)\).
  • Use the x-coordinate to move horizontally along the x-axis.
  • Use the y-coordinate to move vertically along the y-axis.

For example, to plot the point \((-4, -2)\):
1. Begin at the origin.
2. Move 4 units to the left (because the x-coordinate is \(-4)\).
3. Move 2 units down (because the y-coordinate is \(-2)\).

Once you have moved to the correct position, mark it with a dot. Label this dot with the coordinates \((-4, -2)\) to clearly identify it on your graph. Practice with different points to become more comfortable with plotting!

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Most popular questions from this chapter

Change each of the following equations to slope-intercept form, and find the slope. $$ 5 x-y+2=0 $$

What are the slope and the y-intercept of each of these lines? Graph the lines using the slope-intercept method. $$ 3 x+3 y=12 $$

For each of the following, find the slope of the line through the given points. $$ (5,0) \text { and }(0,2) $$

b. Find the equation of the line through each pair of points. $$ (1,1) \text { and }(3,0) $$

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