Slope

The slope of a line characterizes both the steepness and direction of the line.

The slope of a line is the ratio between the change of $y$ and the change of $x$.

Slope is sometimes expressed as rise over run. You can determining slope by visualizing walking up a flight of stairs, dividing the vertical change, which comes first, by the horizontal change, which come second.

The slope, $m,$ of a line is defined to be

$$m = \frac{\text{rise}}{\text{run}} = \frac{\text{change in } y}{\text{change in }x}.$$

Lines that rise from left to right have a positive rise and a positive run, yielding a positive slope.

Lines that fall from left to right have a negative rise and a positive run, yielding a negative slope.

Horizontal lines have zero rise and a positive run, yielding a zero slope.

Vertical lines can have any amount of rise and zero run, yielding an undefined slope.

Line A travels through the points $(-3,4)$ and $(3,-5).$ Line B travels through the points $(-3, 4)$ and $(3, 5).$ Which line has a negative slope?

Answer

Line A has a negative slope because when moving from the lower $x$-value of -3 to the higher $x$-value of 3, it has a horizontal change of positive 6 but a vertical change of -9. Therefore, it has a negative slope.

The slope of a line is the same everywhere on the line. Therefore, when finding the slope of a line from a graph, we can pick any two points on the line to use to find the slope. For the graph below, moving from the point on the left to the point on the right, we move right 2 units and up 3 units. So the ratio of the change in $y$ to the change is $x$ is $\frac{3}{2}.$

What is the slope of the blue line below?

Answer

Choosing two points on the line that travel through lattice points on the grid, we see that moving from left to right the line travels right 2 units and down 7 units. So the ratio of the change in $y$ to the change is $x$ is $\frac{-7}{2}.$

For two different points on a line, $(x_1, y_1)$ and $(x_2, y_2)$, the slope is equal to the ratio $\frac{ y_2 - y_1 } { x_2 - x_1}$. It is also known as the rate of change of $y$ with respect to $x$.

What is the slope of the line through the points $(4,7)$ and $(8,10)\,?$

Answer

The slope is $$\frac{ y_2 - y_1 } { x_2 - x_1} =\frac{10-7}{8-4} = \frac{3}{4}.$$