Slope Calculator

The slope calculator determines the slope or gradient between two points in the Cartesian coordinate system. The slope is basically the amount of slant a line has and can have a positive, negative, zero, or undefined value. Before using the calculator, it is probably worth learning how to find the slope using the slope formula. To find the equation of a line for any given two points that this line passes through, use our slope intercept form calculator.

Here, we will walk you through how to use this calculator, along with an example calculation, to make it simpler for you. To calculate the slope of a line, you need to know any two points on it:

1. Enter the x and y coordinates of the first point on the line.

2. Enter the x and y coordinates of the second point on the line.

3. We instantly get the slope of the line. But the magic doesn't stop there, for you also get a bunch of extra results for good measure:

   • The equation of your function (same as the equation of the line).
   • The y-intercept of the line.
   • The angle the line makes with respect to the x-axis (measure anti-clockwise).
   • Slope as a percentage (percentage grade).
   • The distance between the two points.

For example, say you have a line that passes through the points (1, 5) and (7, 6). Enter the x and y coordinates of the first point, followed by the x and y coordinates of the second one. Instantly, we learn that the line's slope is 0.166667. If we need the line's equation, we also have it now: y = 0.16667x + 4.83333.

You can use this calculator in reverse and find a missing x or y coordinate! For example, consider the line that passes through the point (9, 12) and has a 12% slope. To find the point where the line crosses the y-axis (i.e., x = 0), enter 12% in percent grade (9, 12) as the coordinate of the first point, and x₂ = 0. Right away, the calculator tells us that y₂ = 10.92.

The slope of a line has many significant uses in geometry and calculus. The article below is an excellent introduction to the fundamentals of this topic, and we insist that you give it a read.

Notice that the slope of a line is easily calculated by hand using small, whole number coordinates. The formula becomes increasingly useful as the coordinates take on larger values or decimal values.

If the line on which the points lie is given by the formula

$y = m x + c$

then the slope is $m$ while $c$ is the $y$-intercept.

It is worth mentioning that any horizontal line has a gradient of zero because a horizontal line has the same y-coordinates. This will result in a zero in the numerator of the slope formula. On the other hand, a vertical line will have an undefined slope since the x-coordinates will always be the same. This will result the division by zero error when using the formula.

1. Identify the coordinates $(x_1, y_1)$ and $(x_2, y_2)$. We will use the formula to calculate the slope of the line passing through the points $(3, 8)$ and $(-2, 10)$.

2. Input the values into the formula. This gives us $(10 - 8)/(-2 - 3)$.

3. Subtract the values in parentheses to get $2/(-5)$.

4. Simplify the fraction to get the slope of $-2/5$.

5. Check your result using the slope calculator.

To find the slope of a line, we need two coordinates on the line. Any two coordinates will suffice. We are basically measuring the amount of change of the y-coordinate, often known as the rise, divided by the change of the x-coordinate, known as the run. The calculations in finding the slope are simple and involve nothing more than basic subtraction and division.

Intuitively, the slope of a function captures its rate of change. One of the simplest real-life examples of slope is velocity: the measure of how position changes in time.

Suppose that a cyclist travels along a straight line, and their position at time t with respect to some initial point 0 is given by the formula x(t) = 5t + 3. (Be careful: here, we vary position with time, so x is a function of t, unlike in the previous sections where we varied y with x.) Then, their velocity is v = 5, which is precisely the slope in the formula for x(t). Its physical interpretation is that in a time increment Δt, the cyclist covers an extra distance of 5(Δt).

In the example above, the cyclist was riding with a constant velocity v = 5. But what if, instead, he speeds up with constant acceleration? Suppose the cyclist starts their ride at point x = 0 with zero velocity and acceleration a = 10. This means that starting from 0, the cyclist increases their speed by 10 in every time unit, and so his velocity at time t is v(t) = 10t. Just like velocity is the rate of change of position, acceleration is the rate of change of velocity, so you can read off the value 10 for the acceleration from the velocity formula.

What about the cyclist's position in this case? It turns out that their position is described by the formula x(t) = 5t². We will explain later where this formula comes from: for now, the important point is that in a motion with constant acceleration, position is a quadratic function of time (rather than linear, as it was for the motion with constant speed). As a result, we can no longer read off the cyclist's velocity as a single number from the position formula. This is a consequence of velocity being no longer constant, and it reflects an important lesson: the rate of change (i.e., slope) of a function is a fixed number only if the function is linear.

How, then, can we compute the velocity v(t) = 10t from the position formula x(t) = 5t²? We need to differentiate x(t) (i.e., take its derivative). The derivative of a function at a given time is precisely its rate of change at this time; therefore, v(t) is the derivative of x(t), denoted as x'(t) or dx(t)/dt.

Geometrically, the derivative x'(t) is the slope of the line tangent to the function at time t; in general, at each time t, the tangent line will be different, and so will the derivative. So, differentiation refines the idea of computing slopes, as it allows us to compute the "slope" of functions whose rates of change also vary.

The converse of differentiation is integration: the process in which we compute a function from its derivative. It is through integration that we get the cyclist's position function x(t) = 5t² from their velocity v(t) = 10t. Geometrically, integrating a function amounts to counting the area under the graph. The graph of the velocity function v(t) = 10t determines a triangle with base t and height 10t, and so the area under the graph is 0.5 × t × (10t) = 5t² — that's exactly our position function!

Just as the slope can be calculated using the endpoints of a segment, so can the midpoint. The midpoint is an important concept in geometry, particularly when inscribing a polygon inside another polygon with its vertices touching the midpoint of the sides of the larger polygon. This can be obtained using the midpoint calculator or by simply taking the average of each of the x-coordinates and the average of the y-coordinates to form a new coordinate.

The slopes of lines are important in determining whether or not a triangle is a right triangle. If any two sides of a triangle have slopes that multiply to -1, then the triangle is a right triangle. The computations for this can be done by hand or by using the right triangle calculator. You can also use the distance calculator to compute the longest side of a triangle, which helps determine which sides must form a right angle if the triangle is right.

The sign in front of the gradient provided by the slope calculator indicates whether the line is increasing, decreasing, constant or undefined. If the graph of the line moves from lower left to upper right it is increasing and is therefore positive. If it decreases when moving from the upper left to the lower right, then the gradient is negative.

The slope calculator is one of the oldest at Omni Calculator, built by our veterans Mateusz and Julia, who make creating accurate scientific tools look easy. The idea for this calculator was born when the two were crunching data analytics and trends and realized how a slope calculator would make their job easier. Even today, you can find them occasionally using this tool for reliable calculations.

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