Slope: Four Types and How to Find It | Grade 8 Math | LevelKite

In a proportional relationship $y = kx$ , the constant $k$ tells you how much $y$ changes each time $x$ goes up by $1$ . Its graph is a straight line through the origin $(0, 0)$ .

Many straight lines do not pass through the origin, so they do not show proportional relationships. You can still ask the same question about them: how much does $y$ change for each $1$ unit that $x$ increases? For any straight line that is not vertical, the number that answers this question is the slope of the line.

To measure slope, choose two different points on the line and move from the first point to the second. Compare two changes:

The rise is the change in $y$ . It is positive when $y$ goes up and negative when $y$ goes down.
The run is the change in $x$ . It is positive when you move right and negative when you move left.

The slope is the rise divided by the run, as long as the run is not $0$ . Dividing turns the two changes into a unit rate: the change in $y$ for each $1$ unit of change in $x$ .

Key concept

Slope is rise divided by run

Take two different points $(x_1, y_1)$ and $(x_2, y_2)$ on the same straight line. The slope, written $m$ , is the change in $y$ divided by the change in $x$ . Subtract in the same order on the top and on the bottom.

The division is allowed only when $x_2 \neq x_1$ , because division by $0$ is undefined. If $x_2 = x_1$ , the line is vertical and its slope is undefined.

On a line that is not vertical, every pair of different points gives the same slope. So you may use any two points whose coordinates you know exactly.

m = \frac{y_2 - y_1}{x_2 - x_1}, \quad x_2 \neq x_1

Positive and negative slope. Read a line from left to right, so that $x$ increases and the run is positive. Then the sign of the rise decides the sign of the slope.

If $y$ increases as $x$ increases, the rise is positive. A positive number divided by a positive number is positive, so the line has a positive slope.
If $y$ decreases as $x$ increases, the rise is negative. A negative number divided by a positive number is negative, so the line has a negative slope.

In the figure below, the horizontal axis shows $x$ and the vertical axis shows $y$ . Line A passes through $(-2, -3)$ and $(2, 5)$ . From the first point to the second, both $x$ and $y$ increase, so Line A has a positive slope. Line B passes through $(-2, 4)$ and $(4, -2)$ . Here $x$ increases while $y$ decreases, so Line B has a negative slope.

Line A passes through

(-2, -3)

and

(2, 5)

. Line B passes through

(-2, 4)

and

(4, -2)

.

Zero and undefined slope. A horizontal line has the same $y$ -value at every point. Between any two different points on it, the rise is $0$ and the run is not $0$ . Zero divided by a nonzero number is $0$ , so a horizontal line has a slope of zero. Zero is a number: it tells you that $y$ does not change.

A vertical line has the same $x$ -value at every point. Between any two different points on it, the run is $0$ and the rise is not $0$ . Division by $0$ is undefined, because no number multiplied by $0$ gives a nonzero rise. So a vertical line has an undefined slope: no number describes it. A vertical line is also not the graph of a function $y = f(x)$ , a rule that gives exactly one $y$ -value for each $x$ -value, because its one $x$ -value is paired with many $y$ -values.

In the figure below, $x$ is on the horizontal axis and $y$ is on the vertical axis. Line C passes through $(-4, 3)$ and $(1, 3)$ , so it is horizontal. Line D passes through $(4, -4)$ and $(4, 1)$ , so it is vertical. The table after the figure works out the rise and the run for all four lines.

Line C is horizontal through

(-4, 3)

and

(1, 3)

. Line D is vertical through

(4, -4)

and

(4, 1)

.

Rise, run and slope for Lines A to D, from the first point to the second

Line	First point to second point	Rise (change in y)	Run (change in x)	Slope (rise divided by run)
A	$(-2, -3)$ to $(2, 5)$	$5 - (-3) = 5 + 3 = 8$	$2 - (-2) = 2 + 2 = 4$	$\frac{8}{4} = 2$ , positive
B	$(-2, 4)$ to $(4, -2)$	$-2 - 4 = -6$	$4 - (-2) = 4 + 2 = 6$	$\frac{-6}{6} = -1$ , negative
C	$(-4, 3)$ to $(1, 3)$	$3 - 3 = 0$	$1 - (-4) = 1 + 4 = 5$	$\frac{0}{5} = 0$ , zero
D	$(4, -4)$ to $(4, 1)$	$1 - (-4) = 1 + 4 = 5$	$4 - 4 = 0$	$5 \div 0$ is undefined

Finding slope from a graph. To find a slope from a graph, you need two different points on the line whose coordinates you know exactly, such as marked points that sit on grid intersections. Do not judge a slope by how steep the line looks, because the look of a graph changes with the scale of its axes.

Move from the left point to the right point in two parts. First count the run: how many units you move to the right. Then count the rise: how many units you move up (positive) or down (negative). Finally, divide the rise by the run. On a line that is not vertical, two different points never share an $x$ -value, so the run is not $0$ and the division is allowed.

The figure below shows Line E with three marked points. The horizontal axis shows $x$ and the vertical axis shows $y$ .

Line E passes through

(-2, -1)

,

(0, 2)

and

(2, 5)

.

Worked example

Slope from a graph

Line E in the figure above passes through the marked points $(-2, -1)$ , $(0, 2)$ and $(2, 5)$ . Find the slope of Line E.

Choose two marked points whose coordinates are exact. Take the leftmost point as the first point and the rightmost point as the second, so that you read the line from left to right.
$(-2, -1) \text{ and } (2, 5)$
Count the run. Moving from the first point to the second, you go 4 grid units to the right, so the run is positive 4. Subtracting the x-values, second minus first, gives the same number.
$\text{run} = 2 - (-2) = 2 + 2 = 4$
Count the rise. Over the same move you go 6 grid units up, so the rise is positive 6. Subtracting the y-values in the same order gives the same number.
$\text{rise} = 5 - (-1) = 5 + 1 = 6$
Divide the rise by the run. The run is 4, which is not 0, so the division is allowed. Then divide the top and the bottom by 2, which does not change the value of the fraction.
$m = \frac{\text{rise}}{\text{run}} = \frac{6}{4} = \frac{3}{2}$
Check with a different pair of marked points. From (0, 2) to (2, 5) the run is 2 and the rise is 3. That is half the run and half the rise of the first pair, so the ratio is the same. On a line that is not vertical, every pair of different points gives the same slope.
$m = \frac{5 - 2}{2 - 0} = \frac{3}{2}$

Answer

The slope of Line E is $\frac{3}{2}$ . It is positive: $y$ increases by $3$ for every $2$ units that $x$ increases.

Finding slope from two points. In the last example, counting on the grid and subtracting coordinates gave the same rise and the same run. So you do not need a drawing. When you know the coordinates of two different points on a line, subtract the $y$ -values to get the rise, and subtract the $x$ -values in the same order to get the run. If the run is not $0$ , divide the rise by the run.

Put parentheses around a negative coordinate when you subtract it, so that its sign is not lost.

Worked example

Slope from two points

A line passes through the points $(-3, 4)$ and $(5, -2)$ . Find its slope.

Fix the order of the points before you subtract. Call the point listed first point 1 and the other one point 2. Every subtraction will be point 2 minus point 1.
$(x_1, y_1) = (-3, 4), \quad (x_2, y_2) = (5, -2)$
Subtract the y-values to find the rise. The rise is negative, which tells you that y goes down as you move from point 1 to point 2.
$y_2 - y_1 = -2 - 4 = -6$
Subtract the x-values in the same order to find the run. Subtracting a negative number is the same as adding its opposite.
$x_2 - x_1 = 5 - (-3) = 5 + 3 = 8$
Divide the rise by the run. The run is 8, which is not 0, so the slope is defined. Then divide the top and the bottom by 2, which does not change the value of the fraction.
$m = \frac{-6}{8} = -\frac{3}{4}$

Answer

The slope of the line is $-\frac{3}{4}$ . It is negative: $y$ decreases by $3$ for every $4$ units that $x$ increases.

Keep the subtraction order the same. A common mistake is to subtract the $y$ -values in one order and the $x$ -values in the other. For $(-3, 4)$ and $(5, -2)$ this gives $\frac{-2 - 4}{-3 - 5} = \frac{-6}{-8} = \frac{3}{4}$ , a positive slope. That cannot be right: as $x$ increases from $-3$ to $5$ , $y$ falls from $4$ to $-2$ . The mistake fails because the top measures the change from the first point to the second, while the bottom measures the change from the second point back to the first. The two changes no longer describe the same move, so the answer is the opposite of the true slope. Either order works if you use it on both the top and the bottom: $\frac{4 - (-2)}{-3 - 5} = \frac{6}{-8} = -\frac{3}{4}$ .

Finding slope from a table. Each row of a table of $x$ - and $y$ -values is a point $(x, y)$ , so any two rows give you two points. Find the slope between them exactly as before.

A table has one slope only if its points lie on one straight line. To check, find the slope from each row to the next. If the $x$ -values are all different and every one of these slopes is the same, the points lie on one straight line with that slope. The $x$ -values do not have to go up in equal steps.

The table below describes a tank that is draining. Here $x$ is the time in minutes since the drain opened, and $y$ is the volume of water in the tank in liters.

Water in a draining tank

Time x (minutes)	Volume y (liters)
$2$	$44$
$5$	$35$
$9$	$23$
$12$	$14$

Worked example

Slope from a table

The table above shows the volume of water $y$ , in liters, in a tank $x$ minutes after its drain opens. Check that each row and the next give the same slope. Then state the slope and explain what it means.

Treat the first two rows as two points. Subtract second minus first for the y-values and for the x-values, then divide. The run is 3, which is not 0, so the division is allowed.
$\frac{35 - 44}{5 - 2} = \frac{-9}{3} = -3$
Do the same for the second and third rows. Here x goes up by 4 instead of 3, so y changes by a different amount, but the ratio of the two changes is the same.
$\frac{23 - 35}{9 - 5} = \frac{-12}{4} = -3$
Do the same for the third and fourth rows, to make sure the slope stays the same to the end of the table.
$\frac{14 - 23}{12 - 9} = \frac{-9}{3} = -3$
Compare the three results. They are equal and the x-values are all different, so the four points lie on one straight line. Attach the units: the rise is in liters and the run is in minutes, so the slope is a rate in liters per minute.
$m = -3 \text{ liters per minute}$

Answer

The slope is $-3$ liters per minute. From each row of the table to the next, the volume of water goes down at a rate of $3$ liters per minute.

Takeaway. For a line that is not vertical, slope tells you how much $y$ changes for each $1$ unit that $x$ increases. From a graph, two points or a table, the method is the same: take two different points with exact coordinates, find the rise and the run in the same order, and divide the rise by the run, which must not be $0$ . If $y$ increases as $x$ increases, the slope is positive. If $y$ decreases as $x$ increases, the slope is negative. A horizontal line has a slope of $0$ , and a vertical line has an undefined slope. Next, you will use slope to find a missing coordinate and to graph a line.