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Find the Slope Calculator + Online Solver With Free Steps

The Find the Slope Calculator calculates the slope or gradient of the two-dimensional line joining two points from the coordinates of the points. The coordinates must be two-dimensional (planar).

The calculator supports the Cartesian coordinate system, which can represent both complex and real numbers. Use “i” to depict the imaginary part if your coordinates are complex. Further, note that if you enter variables like x or y, the calculator will simplify and represent the slope in terms of those variables. 

What Is the Find the Slope Calculator?

The Find the Slope Calculator is an online tool that finds the slope/gradient of a line joining any two points – whose coordinates are given – on a two-dimensional plane.

The calculator interface consists of a description of how to operate the calculator and four input text boxes. For your convenience, consider the coordinates of two points:

p1 = (x1, y1)

p2 = (x2, y2) 

Where xk is the abscissa, and yk is the ordinate of the kth coordinate. The calculator requires the values of the abscissa and ordinate for both points separately, and the text boxes are labeled accordingly:

  1. The $\mathbf{y}$ location for the second coordinate: Value of y2
  2. The $\mathbf{y}$ location for the first coordinate: Value of y1.
  3. The $\mathbf{x}$ location for the second coordinate: Value of x2.
  4. The $\mathbf{x}$ location for the first coordinate:  Value of x1.

In your use case, you will have values for x1, x2, y1, and y2 such that:

\[ x_1,\,  x_2 ,\, y_1,\, y_2 \, \in \, \mathbb{{C,\, R}} \]

Where $\mathbb{C}$ represents the set of complex numbers, and $\mathbb{R}$ represents the set of real numbers. Further, the points must be two-dimensional:

\[ p_1,\, p_2 \, \in \, \mathbb{{C^2,\, R^2}} \]

How To Use the Find the Slope Calculator?

You can use the Find the Slope Calculator to find the slope of a line between two points by simply entering the values of the x and y coordinates of the points. For example, suppose you have the following points:

p1 = (10, 5) 

p2 = (20, 8)

Then you can use the calculator to find the slope of the line joining the two points by using the following guidelines:

Step 1

Enter the value of the second point’s vertical coordinate y2. In the example above, this is 8, so we enter “8” without quotes. 

Step 2

Enter the value of the first point’s vertical coordinate y1. For the above example, enter “5” without quotes.

Step 3

Enter the value of the second point’s horizontal coordinate x2. 20 in the example, so we enter “20” without quotes. 

Step 4

Enter the value of the first point’s horizontal coordinate x1. For the example, enter “10” without quotes.

Step 5

Press the Submit button to get the results.

Results

The results contain two sections: “Input,” which displays the input in the ratio form (slope formula) for manual verification, and “Result,” which displays the value of the result itself.

For the example we assumed, the calculator outputs the input (8-5)/(20-10) and the result 3/10 $\approx$ 0.3.

How Does the Find the Slope Calculator Work?

The Find the Slope Calculator works by solving the following equation:

\[ m = \frac{\text{vertical change}}{\text{horizontal change}} = \frac{\text{rise}}{\text{run}} = \frac{y_2-y_1}{x_2-x_1} = \frac{\Delta y}{\Delta x} \tag*{$(1)$} \]

Where m is the slope, (x1, y1) represents the coordinates of the first point, and (x2, y2) are the coordinates of the second point.

Definition

The slope or gradient of a 2D line joining two points, or equivalently two points on a line, is the ratio of the difference between their y (vertical) and x (horizontal) coordinates. This definition of the slope also applies to lines.

Sometimes, the definition is shortened to “the ratio of the rise over the run” or just “rise over run,” where “rise” is the difference in the vertical coordinate and “run” is the difference in the horizontal coordinate. All these shorthands are in equation (1).

The slope can be used to recover the angle of the line joining the two points. Since the angle is only dependent on the ratio and the slope involves the ratio of the difference between y and x coordinates, the angle is:

\[ \tan(\theta) = \frac{\Delta y}{\Delta x} = m \]

\[ \theta = \arctan{m} \]

Gradients of Lines and Curves

When we talk about the slope of a function, if it is a line, then the slope between any two points on the function (line) is the slope of the line between those two points. 

However, on a curve, the slope between any two points changes at different intervals along the curve. Therefore, the slope of a curve is essentially an estimate of the curve’s gradient over an interval. The smaller this interval, the more accurate the value. 

Visually, if the interval on the curve is extremely small, the line represents a tangent to the curve. Thus, in calculus, gradients or slopes of curves at different points are found using the definition of derivatives. Mathematically, if f(x) = y, then:

\[ m = \frac{dy}{dx} = \lim_{x \, \to \, 0} \frac{\Delta y}{\Delta x}  \]

Physical Meaning and Significance of Slope

The term “slope” literally means a rising or falling surface such that one end is at a lower height, and the second is at a greater one. Simply put, the value of slope refers to the steepness of this inclined surface. A road going up a hill is a simple example of such a sloped surface.

The concept of slope is encountered in various branches of mathematics and physics, especially in Calculus. It also forms the basis of machine learning, where the gradient of the loss function guides the machine to its current state of learning, and whether to continue or stop training. 

Sign of Slope

If the slope at a given point on a curve is positive, it means that the curve is currently rising (function value increases as x increases). If the slope is negative, the curve is falling (function value decreases as x increases). Further, the slope of a completely vertical line is $\infty$, while that of a completely horizontal line is 0. 

Solved Examples

Example 1

Consider the two points:

\[ p_1 = (\sqrt{2},\, 49) \qquad p_2 = (4,\, \sqrt{7}) \]

Find the slope of the line joining them.

Solution

Plugging in the values to equation (1):

\[ m = \frac{\sqrt{7}-49}{4-\sqrt{2}} \]

m = -17.92655 

Example 2

Suppose you have the function:

\[ f(x) = 3x^2+2 \]

Find its slope in the interval x = [1, 1.01]. Then find the gradient using the definition of derivatives and compare the results.

Solution

Evaluating the function:

\[ f(1) = 3(1)^2+2 = 5 \]

\[ f(1.01) = 3(1.01)^2+2 = 3.0603+2 = 5.0603 \]

The above serves as our y1 and y2. Finding the slope:

\[ m = \frac{f(1.01)-f(1)}{x_2-x_1} = \frac{0.0603}{0.01} = 6.03\]

Calculating the derivative:

\[ f’(x) = \frac{d}{dx}\,(3x^2+5) = 6x \]

f’(1) = 6(1) = 6

f’(1.01) = 6(1.01) = 6.06 

Our value of 6.03 from the definition of slope is close to these. If we reduced the interval difference $\Delta x = x_2-x_1$ further, then m $\to$ f’(1).

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