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Comprehensive Guide: Coordinate Geometry, Function Graphing & Linear Equations
Understand the Cartesian coordinate system, slope-intercept form (y = mx + b), intercepts, rate of change, and curve analysis.
1. The Cartesian Coordinate Plane & René Descartes
Formulated in 1637 by French philosopher René Descartes, the Cartesian coordinate system bridges algebra and geometry. Any point on a 2D plane is uniquely specified by an ordered pair (x, y) representing perpendicular signed distances from the horizontal x-axis and vertical y-axis intersecting at the origin (0, 0).
2. Linear Functions & The Slope-Intercept Equation
A linear function produces a straight-line graph whose rate of change is constant. Its canonical forms include:
Slope: m = (y₂ − y₁) ÷ (x₂ − x₁) = Δy ÷ Δx
Point-Slope Form: y − y₁ = m(x − x₁)
The variable m defines the slope (steepness and direction), while b is the y-intercept (the coordinates (0, b) where the line crosses the vertical axis).
3. Graduated Step-by-Step Worked Problems
Problem 1: Finding Line Equation from Two Points
Find the slope-intercept equation of the line passing through points A(2, 5) and B(6, 13).
Step 1: Compute slope m:
m = (13 − 5) / (6 − 2) = 8 / 4 = 2.0.
Step 2: Solve for y-intercept b using point (2, 5):
y = mx + b ⟹ 5 = (2)(2) + b ⟹ 5 = 4 + b ⟹ b = 1.
Result: Equation is y = 2x + 1.
Problem 2: Parallel and Perpendicular Lines
For the line y = 2x + 1:
• Any parallel line shares the exact same slope: m_parallel = 2 (e.g. y = 2x − 7).
• Any perpendicular line has a negative reciprocal slope: m_perp = −1 / m = −1/2 = −0.5 (e.g. y = −0.5x + 3).
Problem 3: Intersection Point of Two Linear Functions
Find where line L₁: y = 2x + 1 intersects line L₂: y = −x + 7:
• Set equations equal: 2x + 1 = −x + 7.
• Add x to both sides: 3x + 1 = 7 ⟹ 3x = 6 ⟹ x = 2.
• Substitute x = 2: y = 2(2) + 1 = 5.
Result: The lines intersect at (2, 5).
4. Function Transformations: Shifts, Reflections & Scaling
For any base function y = f(x), transformations follow universal rules:
f(x) + c: Vertical shift upward bycunits.f(x − c): Horizontal shift rightward bycunits.−f(x): Vertical reflection across the x-axis.a · f(x): Vertical stretch by factora(ifa > 1) or compression (if0 < a < 1).
5. Frequently Asked Questions (FAQ)
What is the slope of a vertical line?
A vertical line (equation x = k) has an undefined slope because x₂ − x₁ = 0, resulting in division by zero. A horizontal line (y = c) has a slope of exactly 0.
What is the vertical line test?
A geometric relation is a true mathematical function if and only if no vertical line intersects its graph at more than one point. This ensures every input x maps to exactly one output y.
How do I find the x-intercept of a function?
Set y = 0 and solve the resulting algebraic equation for x. The solutions are called the roots or zeros of the function.
What is the difference between domain and range?
The domain is the complete set of all possible input values (x-values) for which the function is mathematically defined. The range is the complete set of all resulting output values (y-values) produced by the function.