Slope/Gradient of a Line

This article deals with straight lines as graphs of linear functions, i.e. functions of the form $f(x)=m \cdot x+t$.

The m in the equation above is called the slope of the straight line.

The slope of a straight line indicates by how many units the $y$-coordinate of a point changes when its $x$-coordinate changes by one unit. In other words, the slope of a straight line measures how steeply it rises.

Calculating the line slope

The gradient/slope of a straight line can be determined from two different points $P(x_1,y_1)$ and $Q(x_2,y_2)$ , which lie on the straight line, using the difference quotient:

It does not matter which points you choose, the quotient always has the same value.

Example

Determine the slope of the given line. To do this, select two points, for example $P(2 \mid 3)$ and $Q(5 \mid 6)$ as in the sketch. The triangle drawn is called a gradient triangle.

One determines $Δy$ and $Δx$, i.e. the difference of the $y$-coordinates and $x$-coordinates of the given points ...

... plugs in the values for the lengths of $\Delta y$ and $\Delta x$ into the formula.

So the line has the slope:

From the slope to the gradient triangle

Take any two points of the straight lines in the coordinate system and draw lines between them parallel to the coordinate axes. Those lines enclose a right-angled triangle.

From the gradient triangle to the slope

If two points of the straight line are given, a gradient triangle can be drawn between them.

The slope of the straight line is then the length of the vertical cathetus (opposite cathetus) divided by the length of the horizontal cathetus (adjacent cathetus). The slope is positive if the straight line rises and negative if the straight line falls.

This results in the same equation as above:

Perpendicular and parallel lines

Consider two lines with their two line equations

Parallel lines

If

then both lines are parallel.

Perpendicular lines

If

then both lines are perpendicular.

Example:

Example:

$\def\arraystretch{1.25} \begin{array}{l}m_1=1.5; m_2=-\frac{2}{3}\\\Rightarrow m_1\cdot m_2=1.5\cdot\left(-\frac{2}{3}\right)=-1\end{array}$

Angle of inclination

The angle of inclination indicates the angle of a straight line with respect to the $x$-axis. The angle of inclination $\alpha$ of a straight line $y = m \cdot x + t$ satisfies

Slope of special straight lines