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A slope is defined as the steepness of a line. If you consider a ski slope, you can go uphill, downhill or stand flat (horizontal) on the ground.


In mathematical terms, a slope or gradient, is a number denoted to the direction and steepness of a line. It is calculated by finding the ratio of vertical change to horizontal change between any two distinct points of a line. Often expressed by the letter ‘m,’ the alphabetical connotation of the slope goes back as early as the year 1844.


There are four types of slopes; positive, negative, zero/no slope, and undefined. Each is distinguished by the direction it points towards. Below is the detailed explanation on the different types of slopes:


Positive Slope

Positive slopes have inclines towards the right. They go uphill from left to right. Mathematically, the numerator and denominator are both positive in the case of this type of slope.


Some real-life examples of a positive slope include a mountain that has an incline towards the right. Even if you notice a small item such as a cupcake, for example, the frosting mounts upwards from left to right as well.


Negative Slope

Contrastingly, a negative slope occurs when a line goes downhill from left to right. The line will begin at the left and fall to the right. Mathematically, the fraction denoting the steepness of this slope will be negative.


Some real-life examples of a negative slope include a rollercoaster that is about to fall from a height. In this case, the slope of the rollercoaster track is negative since it begins from the left and falls to the right.


Zero or No Slope

This is simply when the line is horizontal. There is no incline or decline in the line, and it remains as flat as it was. Mathematically and true to its name, ‘no slope’ lines will be zero over any number – resulting in the end product being zero as well.


Real-life examples of zero slopes can be the boundary wall or fence around your house. The top along the boundary will remain flat or horizontal at all times. Hence it will have a zero slope.


Undefined Slope

This is the opposite of zero or no slope. It takes the shape of a vertical line that goes up to down or down to up. In fractional terms, the answer is slightly more complicated. This is because any number over zero will give an ‘undefined’ or ‘infinite’ answer that cannot be determined by a calculator.


Some real-life examples include drop tower rides in an amusement part. The tower faces up and is vertically straight, showcasing an undefined slope. Additionally, the legs of your dining table, or any other table for that matter, also have an undefined slope because of their direction.


Steepness of a Slope

The incline, or steepness, of a slope is measured by the absolute value of the slope. The steeper the slope, the greater the absolute value will be. The direction, on the other hand, and as mentioned before, is either increasing, decreasing, horizontal or vertical.


Some important points to remember are:

  • A line is increasing if the slope is positive.
  • A line is decreasing if the slope is negative.
  • A line is horizontal if the slope is nil or zero.
  • A line is vertical if the slope is undefined.


Understanding slopes allows for complex concepts in statistics, calculus and mathematics to be broken down into easier ones. The simple idea of a slope defines and sets the basis for modern technology and the environment around us. As we look around and observe the world we live in, we are surrounded by slopes, and each new building or construction site is built on the different types of slopes that exist.

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