3.1-3.7: Coordinate Geometry Notes:

Coordinates

• Cartesian coordinates are written as: (x, y)

• The horizontal axis is the x-axis, and the vertical axis is the y-axis

• The point (0, 0) is called the origin

• Points are plotted in the form: → x = horizontal distance from origin → y = vertical distance from origin

• Quadrants:

  1. Quadrant I: (+x, +y)

  2. Quadrant II: (−x, +y)

  3. Quadrant III: (−x, −y)

  4. Quadrant IV: (+x, -y)

Drawing Linear Graphs

• A linear graph represents an equation of the form: y = mx + c

• m = gradient (slope)

• c = y-intercept (where the line crosses the y-axis so x=0)

To draw a straight-line graph:

• Create a table of values for x (e.g. −2 to 2)

• Substitute x-values into the equation to find corresponding y-values

• Plot the points (x, y) on a grid

• Connect the points with a straight line

Example: 

Equation: y = 2x − 1 

• • x = −2 → y = −5 

  • x = −1 → y = −3 

  • x = 0 → y = −1 

  • x = 1 → y = 1 

  • x = 2 → y = 3

Gradient of Linear Graphs

• The gradient (m) of a straight line measures its steepness

• Formula: m = (change in y) ÷ (change in x) → m = (y₂ − y₁) ÷ (x₂ − x₁)

• Positive gradient: line rises from left to right

• Negative gradient: line falls from left to right

• Zero gradient: horizontal line

• Undefined gradient: vertical line

• Example: 

  • Points: A(1, 2) and B(4, 8) 

  • Gradient: m = (8 − 2) ÷ (4 − 1) = 6 ÷ 3 = 2

Length and Midpoint of a Line Segment

• To find the length between two points A(x₁, y₁) and B(x₂, y₂), use the distance formula: 

• → Length AB = √[(x₂ − x₁)² + (y₂ − y₁)²]

• To find the midpoint of a line segment AB: → Midpoint M = ((x₁ + x₂) ÷ 2, (y₁ + y₂) ÷ 2)

• Example: Points A(2, 3) and B(6, 7)

  • Length AB = √[(6 − 2)² + (7 − 3)²] = √[16 + 16] = √32 ≈ 5.66

  • Midpoint M = ((2 + 6) ÷ 2, (3 + 7) ÷ 2) = (4, 5)

Equation of a Straight Line

• General form of a straight line: y = mx + c where m = gradient (slope) and c = y-intercept

• You may also encounter:

  • x = k (vertical lines)

  • ax + by = c (standard form)

To find the equation of a line:

• Calculate the gradient: → m = (y₂ − y₁) ÷ (x₂ − x₁)

• Use one point (x₁, y₁) and the gradient in the formula: → y − y₁ = m(x − x₁) → Rearrange to get y = mx + c

1. Example: Points A(1, 2) and B(3, 6)

• • Gradient m = (6 − 2) ÷ (3 − 1) = 4 ÷ 2 = 2

  • Using point A(1, 2): → y − 2 = 2(x − 1) → y = 2x

2. Example: Find the slope and y-intercept of 5x + 4y = 8

• Equation rearranged: 4y = -5x + 8 so y = (-5/4)x + 2

• Gradient m = -5/4 and y-intercept = 2

Parallel Lines

Key Concepts

• Parallel lines have the same gradient (slope)

• They never intersect, no matter how far extended

• If a line has equation y = mx + c, then any line parallel to it will have the same m but a different c

How to Find the Equation of a Parallel Line

• Identify the gradient m from the given line

• Use the gradient with a new point (usually given): → y − y₁ = m(x − x₁) → Rearranged to y = mx + c

• Example:

  • Find the equation of the line parallel to y = 4x − 1 that passes through (1, −3)

  • Gradient = 4

  • Use point (1, −3): → y + 3 = 4(x − 1) → y = 4x − 7

  • Final equation: y = 4x − 7

Perpendicular Lines

Key Concepts

• Perpendicular lines intersect at 90°

• Their gradients are negative reciprocals of each other → If one line has gradient m, the other has gradient −1/m

How to Find the Equation of a Perpendicular Line

• Find the gradient m₁ of the original line

• Calculate the perpendicular gradient: → m₂ = −1/m₁

• Use the point–gradient form with a new point: → y − y₁ = m₂(x − x₁) → Rearranged to y = mx + c

• Example:

  • Find the equation of the line perpendicular to y = 2x + 5 that passes through (4, 1)

  • Original gradient = 2

  • Perpendicular gradient = −1/2

  • Use point (4, 1): → y − 1 = −½(x − 4) → y = −½x + 3

NOTES DONE BY FARIDA SABET

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