Functions – Finding Slope

While learning geometry, we come across so many lines, like perpendicular, vertical, horizontal, parallel, etc. How can we find the steepness? How steep can these lines go? There comes the use of slope. The slope gives us an idea of how vertical the line is, or what direction a line goes in a geometric plane. Finding slope is not a tough task. There are so many methods to determine the slope of a line. Here, in this article, let’s see the definition of slope, its meaning, and finding slope using different methods. 

What is Slope?

The slope is the measure of the steepness of a line in a geometric plane. It is determined by finding the ratio between the vertical and the horizontal axis. Or, in simple terms, a Slope is the ratio between the change in the x-axis and the y-axis. It is represented by the letter ‘m’ in Mathematics. With the help of slope, we can identify whether a line is a perpendicular, parallel, horizontal or vertical line. Another important feature of the slope is it remains constant at every point of the line.

Finding Slope

There are different methods used in finding the slope of a line. It may vary in different contexts. 

1. Standard Form: If we want to find the slope of a line with equation ax+ by = c, then the slope is m = – (a / b). This is the most common method used in finding slope. 

2. Slope Intercept Form: The slope-intercept form is a linear equation; y = mx + c; where m is the slope, and c is the y-intercept.

3. Point intercept form: This method is used when points of a line are given. m= ∆y/ ∆x

Varieties of Slope

There are four categories of a slope.

1. Positive Slope – Any line with a positive slope is inclined upwards from left to right. There is a rise when the line is represented. That is, when x increases, y also increases and vice versa. If m is greater than 0, the line has a positive slope by using the equation. 

2. Negative Slope – Any line with a negative slope is inclined downwards from left to right. There is a fall in the line when it is represented. As x increases, y decreases. In this case, m is less than 0. 

3. Zero Slope – There is no rise while representing the line in the coordinate plane. Thus, on calculating the slope, we get zero. The slope is termed Zero slope as the value of the slope is zero. 

4. Undefined Slope – This is the most complicated category of a slope. In this case, the slope is undefined. There is no change in x no matter how high or low the y value goes. 

Slope Equation

The slope is represented by m. The change in x coordinate is represented by ∆x, and the y coordinate is represented by ∆y. The slope is the change in the y coordinate concerning the x coordinate. So, the equation of slope is 

     m = ∆y/∆x

Another representation of this equation is 

    Tan θ = ∆y/∆x

  Let (x1, y1) and (x2, y2) be the points of a straight line in a plane. 

Hence, the slope is 

  m = (y2 – y1)/ (x2 – x1)

Example: If a line passes through points A (10, 12) and B (14, 20), calculate the slope.

      Solution: x1 = 10, x2 = 14, y11 = 12, y2 = 20

                 So, slope m = (y2 – y1) / (x2 – x1) 

                                        = (20 – 12) / (14 – 10)

                                        = 8/4

                                        = 2 

As the slope is greater than 0, it is a positive slope.

The Slope of Various Geometric Elements 

Geometric lines like horizontal, parallel, vertical, and perpendicular lines have different slopes based on their properties on a geometric plane. 

• Horizontal line – A horizontal line is always parallel to the x-axis. It can either be from left to right or right to left in a plane. In horizontal lines, there is no change in y coordinates. Hence, the slope of a horizontal line is zero.

• Vertical line – A vertical line is parallel to the y – axis. It is drawn from top to bottom or bottom to top in a coordinate plane. In this case, there is no change in y coordinates. Hence, the slope of a vertical line is undefined. 

• Perpendicular Line – Perpendicular lines have a 90- degree angle between them. Imagine we have two perpendicular lines, L1 and L2. They are inclined at an angle ∅1 and ∅2. According to the External angle Theorem, ∅2 = ∅1 – 90° 

            Therefore, slope of L1 is m1 = tan ∅1      m2 = tan (∅1 – 90°) = – cot ∅1

                 m1*m2 = -1

                  Hence the slope is -1

• Parallel Line – In parallel lines, the inclination angle is always equal. As the angles of inclinations are equal, the slope of two parallel lines is also equal. 

Conclusion

Finding Slope is an integral part of geometry. On a line, we will be aware of almost all the properties of a line in a geometrical plane. There are 4 types of slopes, positive, negative, zero, and undefined. The slope value determines what type of line is passing through the plane.