3. Determining the slope of a line

Determining the slope of a line

The slope of a line is the ratio of the change in the [[$ y $]]-coordinates to the corresponding change in the [[$ x $]]-coordinate. The slope describes the steepness of the line.

The slope of a line tells how much the [[$ y $]]-coordinate changes (increases or decreases) when the [[$ x $]]-coordinate increases by one. The slope can be calculated by using two points of a line. The slope is usually denoted by the letter [[$ k $]].

The slope [[$k$]] of a line passing through the points [[$ x_1 $]], [[$ y_1 $]] and [[$ x_2 $]], [[$ y_2 $]] is
[[$$ k = \displaystyle\frac {y \text {-axis change }}{x \text {-axis cange}} = \displaystyle\frac {y_2 – y_1} {x_2 – x_1}. $$]]

The effect of the slope on a straight line graph

If [[$ k > 0 $]], the line is ascending.
If [[$ k < 0 $]], the line is descending.
If [[$ k = 0 $]], the line is parallel to the [[$ x $]]-axis.
If [[$ k $]] is undefined, the line is parallel to the [[$ y $]]-axis.

Example 1

Determine the slope of the line in the image.

The slope of an ascending line is always positive. As the [[$ x $]]-coordinate of a line increases by one, the [[$ y $]]-coordinate increases by two. For example, moving one step forward from [[$ (1, 1) $]] will take you straight to [[$ (1 + 1, 1 + 2) = (2, 3) $]].

Example 2

Determine the slope of the line in the image.

The slope of a descending line is always negative. As the [[$ x $]]-coordinate of the line increases by one, the [[$ y $]]-coordinate decreases by one. For example, moving one step forward from [[$ (0, 4) $]] will take you straight to the point [[$ (0 + 1, 4 – \displaystyle\frac {3} {4}) = (1, 3\displaystyle\frac {3} {4}) $]].

Example 3

The line passes through [[$ (1, -4) $]] and [[$ (-2, 5) $]]. Examine by calculating whether the line is ascending or descending.

Determine the slope of the line by using the given points.

Answer: The slope is negative, which means that the line is descending.

NB! Be careful when calculating the coordinates of the points. The [[$ y $]]-coordinate of the point you put first in the formula must also be the first [[$ x $]]-coordinate.

The slope unit

The unit of a line's slope is obtained from the units of the horizontal and vertical axes.

[[$$ \text {The slope unit} = \displaystyle\frac {\text {vertical axis unit}} {\text {horizontal axis unit}} $$]]