The simplest function that still tells a story
A linear function f(x) = ax + b produces a straight line on any graph. Two numbers - the slope a and the y-intercept b - are enough to pin down every property of that line: where it crosses both axes, how steep it is, and whether it climbs or falls. This calculator extracts all four key facts from those two coefficients.
Quick start
Enter a = 3, b = -6. Result: f(x) = 3x + (-6). Zero at x = 2. Y-intercept at (0, -6). The function is increasing because a > 0.
Key formulas
Equation: f(x) = ax + b (slope-intercept form)
X-intercept (zero): x0 = -b / a (solve ax + b = 0)
Y-intercept: (0, b) - just read coefficient b
Slope: a > 0 = increasing, a < 0 = decreasing
Slope reference table
| Slope (a) |
Meaning |
Real-world analogy |
| a = 0.5 |
Gentle rise |
Wheelchair ramp (1:12 ratio = a = 0.08) |
| a = 1 |
45-degree angle |
1 unit up for every 1 unit right |
| a = 3 |
Steep rise |
Aggressive stock price trend |
| a = -1 |
45-degree downhill |
Phone battery draining steadily |
| a = -0.1 |
Gentle decline |
Slow depreciation of equipment |
Practical examples
Taxi fare: Base fare $3.50, plus $2.10 per mile. f(x) = 2.10x + 3.50. Zero at x = -1.67 (no practical meaning - you can't drive negative miles). At 5 miles: f(5) = $14.00. The slope $2.10/mile is the rate you care about.
The slope is the per-unit cost, b is the fixed cost
Temperature conversion: Celsius to Fahrenheit: F(C) = 1.8C + 32. Zero at C = -17.78 (where Fahrenheit also gives a round negative number). Y-intercept: 32 (freezing point of water in Fahrenheit).
The classic linear function from physics class
Depreciation: A laptop worth $1200 loses $200/year in value. f(t) = -200t + 1200. Zero at t = 6 years (fully depreciated). After 3 years: f(3) = $600. Negative slope = decreasing value over time.
Straight-line depreciation is literally a linear function
Water tank: A tank drains at 15 liters/minute from an initial 450 liters. f(t) = -15t + 450. Zero at t = 30 minutes (tank empty). After 10 min: 300 liters remain.
The zero tells you when the resource runs out
Break-even analysis: Revenue R(x) = 25x, Cost C(x) = 15x + 500. Profit P(x) = R - C = 10x - 500. Zero at x = 50 units. You need to sell 50 units to break even. Below 50: loss. Above 50: profit.
The x-intercept is the break-even point
Linear vs quadratic - when to use which
| Property |
Linear f(x) = ax + b |
Quadratic f(x) = ax2 + bx + c |
| Graph shape |
Straight line |
Parabola (curve) |
| Rate of change |
Constant |
Varies with x |
| Max zeros |
1 |
2 |
| Has vertex |
No |
Yes (min or max) |
| Use case |
Constant rate (taxi, depreciation) |
Acceleration, area, optimization |
FAQ
What if a = 0?
When a = 0, f(x) = b for all x - a horizontal line with no slope. This is a constant function, not a linear function. It has no x-intercept (unless b = 0, in which case it lies on the x-axis entirely). The calculator returns an error because the zero formula -b/a would divide by zero.
What is the difference between slope and rate of change?
For a linear function, they are the same thing. The slope a tells you how much f(x) changes when x increases by 1. If a = 3, every unit increase in x adds 3 to f(x). In calculus, the derivative of ax + b is simply a - the slope is constant everywhere on the line.
How do I find the slope from two points?
Given points (x1, y1) and (x2, y2), the slope is a = (y2 - y1) / (x2 - x1). Then b = y1 - a*x1. Example: points (1, 5) and (3, 11). Slope = (11 - 5)/(3 - 1) = 6/2 = 3. Then b = 5 - 3*1 = 2. So f(x) = 3x + 2.
Can a linear function have no zero?
Every linear function with a non-zero slope crosses the x-axis exactly once. The zero always exists at x = -b/a. The only case with "no zero" is the constant function f(x) = b (a = 0, b non-zero), which is a horizontal line that never touches the x-axis. But that is not a linear function by the strict definition.
What does f(x) = mx + b look like in other notations?
The same line can be written as: slope-intercept y = mx + b, standard form Ax + By = C, point-slope y - y1 = m(x - x1), or parametric (x, y) = (x1, y1) + t(1, m). All describe the same line. This calculator uses slope-intercept form because it directly reveals the slope and y-intercept.
How is this related to systems of linear equations?
A system of two linear equations asks: where do two lines intersect? If the slopes differ (a1 != a2), there is exactly one intersection point. If slopes are equal but intercepts differ, the lines are parallel (no solution). If both are equal, the lines overlap (infinite solutions). Solving the system means finding the x where both functions give the same y.
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