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Chapter 1: Problem 4

In Exercises 1鈥�12, plot the given point in a rectangular coordinate system. $$(-1,4)$$

Short Answer

Expert verified
The point (-1,4) can be plotted by moving one step to the left and four steps upwards on the cartesian coordinate system starting from the origin.

Step by step solution

01

Understanding the Cartesian Coordinate System

The Cartesian Coordinate System is a two-dimensional system that is defined by two perpendicular axes. The horizontal axis is often labeled as 'x' and the vertical as 'y'. At the intersection of these axes, the point (0,0) is found, also called the origin.
02

Identify the Given Point in the Coordinate System

The point given is (-1,4). The first number represents the position on the x-axis and the second number represents the position on the y-axis. So, from the origin, move one step to the left (as it's negative) on the x-axis and four steps upwards on the y-axis, which will lead to the required point.
03

Plot the Point

After moving 'one step to the left' and 'four steps up' from the origin, plot the point at this position. This point is (-1,4) on the 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 correctly is a fundamental skill in coordinate geometry. When you receive a point like \((-1, 4)\), it represents a location on a Cartesian coordinate system. To plot this point:
  • Start at the origin, which is the point \((0, 0)\).
  • The first number, \(-1\), indicates you need to move left along the x-axis since it is negative.
  • The second number, \(4\), means moving up 4 units along the y-axis.
  • Place a dot where these movements intersect. This marks \((-1, 4)\).
Plotting points requires attention to detail, ensuring each movement is exact for accurate representation.
Rectangular Coordinate System
The rectangular coordinate system, often called the Cartesian plane, consists of a flat surface defined by two lines perpendicular to each other. These lines are called axes. This system is a framework to plot points and visualize the relationship between numbers.The x-axis runs horizontally, while the y-axis runs vertically. The two axes meet at the "origin," denoting the starting point of \((0, 0)\). The rectangular coordinate system is divided into four sections, known as quadrants:
  • Quadrant I: where both x and y are positive
  • Quadrant II: where x is negative and y is positive
  • Quadrant III: where both x and y are negative
  • Quadrant IV: where x is positive and y is negative
Knowing each quadrant's characteristics helps understand where certain points will lie after plotting.
Axes and Origin
The axes in a rectangular coordinate system serve as reference lines for plotting and calculating points. Here's what each component signifies:
  • x-axis: A horizontal line used for measuring the horizontal distance from the origin. Positive values extend to the right, and negative values extend to the left.
  • y-axis: A vertical line used for measuring vertical distance. Positive values go upwards, while negative values go downwards.
  • Origin: The point \((0, 0)\), where the x-axis and y-axis intersect. It acts as a central point for reference when plotting points.
Understanding axes and the origin is crucial for navigating and using the Cartesian coordinate system effectively.
Coordinate Geometry
Coordinate geometry, also known as analytic geometry, studies geometric figures using coordinates and algebra. This approach allows the representation of points, lines, and shapes on a plane using equations. Key features of coordinate geometry include:
  • Analysis of Shapes: By converting geometric shapes into algebraic equations, one can easily analyze properties like distance, midpoints, and slopes.
  • Finding Relationships: The position and movement of points can reveal relationships between lines, as well as between points and lines.
  • Simplicity in Calculation: With formulas like the distance formula or slope, coordinate geometry simplifies many problems encountered in classic geometry.
By learning coordinate geometry, students can connect algebraic and geometric insights, providing a thorough understanding of both mathematical realms.

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