Slopes and Gradients - Revision Notes

In these revision notes for Slopes and Gradients, we cover the following key points:

• What is a slope?
• What is the gradient of a line?
• What is the key difference between the concepts 'slope' and 'gradient'?
• What does the gradient of a line indicate on a graph?
• How do we calculate the gradient of a line?
• What does the sign of a gradient tell us about the line?
• What is the gradient of a horizontal/vertical line?

Slopes and Gradients Revision Notes

In mathematics, when using the word 'slope' we commonly refer to a certain inclination that causes a motion, a line or a surface to diverge from the vertical or horizontal direction. Unlike in everyday life, where this term is associated with the inclination of a surface, the term 'slope' in math is mostly used to describe the overall feature of a graph without going into numerical details. In a certain sense, the term 'slope' is a synonym for 'steepness'.

The term 'gradient' is more number-oriented. It describes the rate of inclination in the sense that we have to calculate how much the vertical shift changes with respect to the horizontal shift when the position of a mathematical object (it can be simply a point) changes.

When dealing with a 2-D graph (i.e. an X-Y graph), we make all calculations based on the origin of the coordinate system, which we consider as a reference point. In this sense, when giving the coordinates of a certain point, we tell how far from the origin this point is in the horizontal and vertical directions.

However, a line is not made of a single point but of an infinite number of points instead. Therefore, all graphs - linear and otherwise - contain an infinity of points that change their coordinates gradually when shifting throughout that line.

All points in the same straight line are collinear. Since the line has the same steepness everywhere, we expect the gradient of this line to be the same number.

By definition, the gradient k (or m) of a line represents the change in the vertical coordinate divided by the change in the horizontal coordinate.

When calculating the gradient of a line, we usually consider two known points and use their coordinates to find the value of the gradient.

In symbols, we write

where x₁ and y₁ are the coordinates of the leftmost point while x₂ and y₂ are the coordinates of the rightmost point considered. In linear graphs (straight lines), we can use any two points to calculate the gradient as the result will always be the same for any pair considered.

The gradient of a linear graph shows how much faster the y-coordinates of any two points of that graph change compared to the corresponding x-coordinates. In the theory of functions, the gradient of a graph line is known as the "rate of a function's change."

The sign of the gradient provides very important information in regard to the slope. We have the following cases when dealing with gradients:

1. If the gradient is a positive finite number, the line increases in the quote (the line is increasing) when shifting from left to right. We can distinguish three sub-cases related to this situation:
   If the gradient is greater than 1, the y-coordinates increase faster than the corresponding x-coordinates;
   1. If the gradient is equal to 1, the y-coordinates increase at the same rate as the corresponding x-coordinates; and
   2. If the gradient is smaller than 1, the y-coordinates increase slower than the corresponding x-coordinates.
2. If the gradient is negative, the line decreases in the quote (is decreasing) when shifting from left to right. Here again, we may deal with three different situations:
   1. If the gradient is smaller than -1, then the y-coordinates decrease at a higher rate than the increase in the corresponding x-coordinates;
   2. When the y-coordinate of a line decreases at a slower rate than the increase in the corresponding x-coordinate, we have a negative gradient greater than -1; and
   3. If the gradient is equal to -1, any increase in the x-coordinate brings a decrease in the same rate of the corresponding y-coordinate.

When dealing with the gradient of a line, the change in the x-coordinate is always taken as positive as we always consider movements from left to right. The sign of the gradient is therefore determined by the changes in the corresponding y-coordinate (moving up = positive, moving down = negative).

If a line is horizontal, this means there is no change in the vertical coordinate for whatever change in the horizontal coordinate. In symbols, we have Δy = 0 for all values of Δx ≠ 0. From the formula of gradient, we therefore obtain

On the other hand, the gradient of vertical lines is considered as equal to infinity, because there is no change in the x-coordinate (Δx = 0) for whatever change in the y-coordinate (Δy ≠ 0). In mathematics, a number divided by zero is undetermined, but if we think of it as a non-zero number divided by a very small number, getting closer to zero, the value of the fraction obtained approaches infinity. In symbols, we have