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Line Graph (y = mx + b) – Tutorial
On this page, you can find the logic, usage, and important details of the Line Graph (y = mx + b) calculator.
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The Line Equation: y = mx + b
y = mx + b is the most common form for a straight line in the coordinate plane. Just two parameters (m and b) fully determine the line.
Slope (m): Direction of the Line
m is the slope: how much y changes when x increases by 1.
m = Δy / Δx = (y₂ − y₁) / (x₂ − x₁)
m > 0 → line goes up · m < 0 → line goes down · m = 0 → horizontal (y = b)
Larger |m| means a steeper line.
Larger |m| means a steeper line.
Intuition: "Move right 1 — how much does y change?" m = 2: right 1 → up 2. m = −3: right 1 → down 3.
Y-intercept (b): Where the Line Meets the y-axis
b is the y value where the line crosses the y-axis. Setting x = 0:
y(0) = m·0 + b = b → crossing point: (0, b)
b > 0 → crosses above origin · b = 0 → passes through origin · b < 0 → crosses below
X-intercept: Where the Line Meets the x-axis
Found by setting y = 0:
0 = mx + b → x = −b / m (m ≠ 0)
If m = 0, the line is horizontal; if b ≠ 0 it never crosses the x-axis.
Finding the Line Equation
Given slope + one point
Point (x₀, y₀) and slope m known:
y − y₀ = m(x − x₀) → b = y₀ − m·x₀
y − y₀ = m(x − x₀) → b = y₀ − m·x₀
Given two points
m = (y₂ − y₁) / (x₂ − x₁)
b = y₁ − m·x₁
b = y₁ − m·x₁
Parallel and Perpendicular Lines
| Relationship | Condition | Example |
|---|---|---|
| Parallel | m₁ = m₂, b₁ ≠ b₂ | y = 2x+1 and y = 2x−3 |
| Perpendicular | m₁ · m₂ = −1 | y = 2x and y = −½x |
| Same line | m₁ = m₂, b₁ = b₂ | y = 3x+2 and y = 3x+2 |
How to Read the Graph
- Point (0, b) always lies on the line — find it where the line crosses the y-axis.
- To feel the slope m: "If I move x right by 1, how much does y change?"
- To find the x-intercept: look for y = 0 in the table, or use x = −b/m.
- Reducing the table step shows more points; increasing it shows fewer.
Real-World Applications
- Pricing: "Base fee b + rate-per-unit m" → linear cost model
- Physics: Constant-velocity motion: position = speed · time + initial position
- Economics: Simple supply–demand and cost models
- Unit conversion: Temperature: F = 1.8·C + 32 (m = 1.8, b = 32)
Note: y = mx + b cannot represent vertical lines (x = constant); vertical lines have undefined slope.