MathematicsChapter 7: Coordinate Geometry
The Cartesian Plane
The Cartesian plane is a grid made from two number lines that cross at right angles.
Learning objectives
- Explain the cartesian plane in clear mathematical language
- Apply the ideas in ordered pairs
- Solve problems involving the axes
Definition
The Cartesian plane is a grid made from two number lines that cross at right angles. A point is an ordered pair (x,y): x is the horizontal coordinate and y is the vertical coordinate. The origin (0,0) is where the axes meet. On the x-axis y=0. On the y-axis x=0. Ordered matters: (x,y)=(y,x) if and only if x=y. Otherwise (2,5) is not the same point as (5,2).
Mathematical form
y=0. On the y-axis x
x=y. Otherwise (2
Where / symbols
- ordered pair (x,y)
- A point is an ordered pair (x,y): x is the horizontal coordinate and y is the vertical coordinate.
- origin (0,0)
- The origin (0,0) is where the axes meet.
- x-axis y=0
- On the x-axis y=0.
- y-axis x=0
- On the y-axis x=0.
Relationship
- x-axis y=0
- y-axis x=0
- (x,y)=(y,x)
- x=y
Overview
Name the point, then name the axes:
Ordered Pairs
An ordered pair (x,y) lists run first and rise second. The origin (0,0) is the zero of both axes. To plot (3,-2), move 3 right and 2 down. Because order matters, (x,y)=(y,x) iff x=y: only points on the line y=x have equal coordinates swapped. (4,4) swapped is still (4,4); (4,1) swapped is a different point.
Examples (0,0) the origin · (3,-2) right 3, down 2 · (4,4) equals its swap
The Axes
The x-axis is the horizontal line y=0. The y-axis is the vertical line x=0. Every point on the x-axis looks like (a, 0). Every point on the y-axis looks like (0, b). The axes divide the plane into four regions called quadrants, which the next page names by sign.
Examples (5,0) on the x-axis · (0,-3) on the y-axis · (0,0) on both axes
What it means
The Cartesian plane is a grid made from two number lines that cross at right angles.
Main ideas
1. ORDERED PAIRS: An ordered pair (x,y) lists run first and rise second. 2. THE AXES: The x-axis is the horizontal line y=0.
Why it matters
Descartes linked algebra to geometry by this grid: an equation in x and y becomes a picture, and a picture becomes coordinates. Always write the pair in round brackets with x first. A missing sign on y will throw the point into the wrong half-plane.
How to answer in an exam
(x,y)=(y,x) only when x=y. Do not call (2,5) the same as (5,2). Origin is (0,0), not empty paper. On the x-axis y=0; on the y-axis x=0. A point with both coordinates nonzero is off the axes.
Explanation
Descartes linked algebra to geometry by this grid: an equation in x and y becomes a picture, and a picture becomes coordinates. Always write the pair in round brackets with x first. A missing sign on y will throw the point into the wrong half-plane.
In simple words
(x,y)=(y,x) only when x=y. Do not call (2,5) the same as (5,2). Origin is (0,0), not empty paper. On the x-axis y=0; on the y-axis x=0. A point with both coordinates nonzero is off the axes.
Key points
- ordered pair (x,y)
- origin (0,0)
- x-axis y=0
- y-axis x=0
Exam preparation
Exam Focus
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.The Cartesian Plane
Sticky Notes · Quick Revision
Core Idea
- The Cartesian plane is a grid made from two number lines that cross at right angles.
Key Terms
- ordered pair (x,y): A point is an ordered pair (x,y): x is the horizontal coordinate and y is the vertical coordinate.
- origin (0,0): The origin (0,0) is where the axes meet.
Main Pattern
- 1. ORDERED PAIRS: An ordered pair (x,y) lists run first and rise second.
- 2. THE AXES: The x-axis is the horizontal line y=0.
Worked Reminder
- (0,0) the origin
- (3,-2) right 3, down 2
- (4,4) equals its swap
Common Mistake
- (x,y)=(y,x) only when x=y. Do not call (2,5) the same as (5,2). Origin is (0,0), not empty paper. On the x-axis y=0; on the y-axis x=0. A point with both coordinates nonzero is off the axes.
Exam Tip
- State the rule for the cartesian plane, show each algebraic step, and check the final answer against the question.
60-second revision
- The Cartesian plane is a grid made from two number lines that cross at right angles.
- An ordered pair (x,y) lists run first and rise second.
- The x-axis is the horizontal line y=0.
- Descartes linked algebra to geometry by this grid: an equation in x and y becomes a picture, and a picture becomes coordinates.
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