Cartesian coordinates form the backbone of the coordinate geometry, allowing us to navigate and plot points on a 2D plane with precision. This system enables a clear visual representation of mathematical concepts and is vital for a variety of applications in both mathematics and the sciences.

Cartesian Coordinate System

Introduced by René Descartes, the Cartesian coordinate system features two axes: the x-axis (horizontal) and the y-axis (vertical), intersecting at the origin (0, 0). This simple yet powerful system can describe any point in 2D space using a pair of numbers.

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Components of Cartesian Coordinates

• X-coordinate (abscissa): Horizontal position from the origin.
• Y-coordinate (ordinate): Vertical position from the origin.

Plotting Points on the Cartesian Plane

To plot a point, you move from the origin to the x-coordinate on the x-axis, then parallel to the y-axis to reach the y-coordinate.

Example 1: Plotting a Point

Objective: Plot $A(3, 2)$.

1. Move 3 units right (x-axis).

2. Move 2 units up (y-axis).

3. Mark the point $A$.

The point $A(3, 2)$ is now plotted.

Interpreting Points on the Cartesian Plane

Every point's coordinates offer insight into its position relative to other points or features on the plane.

Example 2: Distance Between Points

Find the distance between $A(1, 2)$ and $B(4, 6)$.

Solution:

1. Distance formula:

2. Apply coordinates:

3. Calculate:

The distance between $A$ and $B$ is 5 units.

Quadrants and Axes

The plane is divided into four quadrants by the x and y axes, with each quadrant representing a unique combination of positive and negative values for x and y.

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Example 3: Identifying Quadrants

Objective: Determine the quadrant for $C(-5, 3)$.

Solution:

• Negative x and positive y: Quadrant II.

Functions and Graphs

Cartesian coordinates are essential for graphing functions and analyzing their properties.

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Example 4: Plotting a Linear Function

Plot $y = 2x + 1$ for $x = -2, -1, 0, 1, 2$.

Solution:

1. Calculate $y$ for each $x$.

• For $x = -2$, $y = 2(-2) + 1 = -3$
• For $x = -1$, $y = 2(-1) + 1 = -1$
• For $x = 0$, $y = 2(0) + 1 = 1$
• For $x = 1$, $y = 2(1) + 1 = 3$
• For $x = 2$, $y = 2(2) + 1 = 5$

2. Plot and connect the points.

Reflections and Transformations

Transformations such as reflections are straightforward to execute within the Cartesian system.

Example 5: Reflecting a Point Across the Y-axis

Reflect $D(4, -3)$ across the y-axis.

Solution:

• Reflection changes $x$ to $-x$: $D'(-4, -3)$.

Practical Applications

The Cartesian coordinate system finds applications in numerous fields, from engineering to computer graphics, illustrating its importance beyond the classroom.