📈 Gradient Calculator
Calculate the gradient (slope) between two points
Point 1 (x₁, y₁)
Point 2 (x₂, y₂)
How to Use This Calculator
Enter Two Points
Input the coordinates of two points: (x₁, y₁) and (x₂, y₂) on the line.
Click Calculate
Press "Calculate Gradient" to find the slope between the two points.
View Result
See the gradient (slope) displayed, along with the angle the line makes with the x-axis.
Formula
Gradient = (y₂ - y₁) / (x₂ - x₁) = Δy / Δx
Also known as: slope, rise over run, rate of change
Where:
- (x₁, y₁) = First point
- (x₂, y₂) = Second point
- Δy = Change in y (y₂ - y₁)
- Δx = Change in x (x₂ - x₁)
Example: Points (1, 2) and (3, 6)
Gradient = (6 - 2) / (3 - 1) = 4 / 2 = 2
The line rises 2 units for every 1 unit it runs to the right.
About Gradient Calculator
The Gradient Calculator finds the gradient (slope) of a line connecting two points. The gradient measures how steep a line is and represents the rate of change of y with respect to x.
When to Use This Calculator
- Mathematics: Calculate slopes in coordinate geometry
- Physics: Find velocity, acceleration, and other rates of change
- Engineering: Analyze gradients in construction, roads, and slopes
- Economics: Calculate rates of change in cost, revenue, or profit
- Graphing: Understand line steepness and direction
- Education: Learn and practice gradient/slope concepts
Why Use Our Calculator?
- ✅ Simple Formula: Uses the standard gradient formula
- ✅ Angle Calculation: Shows the angle the line makes with the x-axis
- ✅ Step-by-Step Display: Shows the calculation process
- ✅ Works with All Numbers: Handles decimals, fractions, and negatives
- ✅ 100% Accurate: Precise mathematical calculations
- ✅ Completely Free: No registration required
Understanding Gradient
The gradient (or slope) is the ratio of vertical change (rise) to horizontal change (run) between two points on a line. It describes how steep the line is and in which direction it slopes.
- Positive gradient: Line slopes upward (left to right)
- Negative gradient: Line slopes downward (left to right)
- Zero gradient: Horizontal line (no change in y)
- Undefined gradient: Vertical line (x₂ = x₁, no change in x)
- Larger absolute value: Steeper line
- Gradient = tan(angle) where angle is measured from x-axis
Real-World Applications
Construction: Road gradients, roof slopes, and ramps use gradient calculations to ensure proper drainage and accessibility.
Physics: Velocity-time graphs use gradients to find acceleration. Position-time graphs use gradients for velocity.
Economics: Supply and demand curves use gradients to measure elasticity and responsiveness to price changes.
Frequently Asked Questions
What is a gradient?
Gradient (or slope) is the ratio of vertical change to horizontal change: gradient = (y₂ - y₁) / (x₂ - x₁) = Δy / Δx. It measures how steep a line is.
What's the difference between gradient and slope?
Gradient and slope are the same thing! "Gradient" is more common in UK/Australia, while "slope" is more common in US. Both mean the same mathematical concept.
Can gradient be negative?
Yes! A negative gradient means the line slopes downward as you move from left to right. For example, gradient = -2 means the line falls 2 units for every 1 unit to the right.
What if x₂ equals x₁?
If x₂ = x₁, the line is vertical and the gradient is undefined (division by zero). Vertical lines have no gradient in the traditional sense.
What's the relationship between gradient and angle?
Gradient = tan(θ), where θ is the angle the line makes with the positive x-axis. So angle = arctan(gradient).
How is gradient used in calculus?
Gradient is the derivative of a function at a point. For linear functions, the gradient is constant. For curves, the gradient at each point gives the slope of the tangent line.
What does "rise over run" mean?
"Rise over run" is another way to describe gradient: rise (vertical change, Δy) divided by run (horizontal change, Δx). It's a simple way to visualize slope.