Math Core

Some quantities change at a steady pace: a phone plan adds the same charge every month, a hiker climbs the same number of feet every hour. When a quantity changes at a constant rate, its graph is a straight line, and the number that measures that rate is the line's slope. Slope is the single most important idea in this unit.

Rate of change

Suppose a pool is being filled with a hose. You record the water level every $2$ minutes.

time (min), $x$	$0$	$2$	$4$	$6$
depth (in.), $y$	$10$	$13$	$16$	$19$

Every time $x$ goes up by $2$ , $y$ goes up by $3$ . So the depth rises $\dfrac{3}{2} = 1.5$ inches per minute. That number is the rate of change:

\text{rate of change} = \frac{\text{change in } y}{\text{change in } x}.

Because the rate is the same between every pair of rows, this relationship is linear. In the last unit you saw the same pattern in arithmetic sequences: a constant difference. Slope is that constant difference, measured per one unit of $x$ .

Slope on a graph

On a graph, the change in $y$ is how far you move up or down, called the rise. The change in $x$ is how far you move left or right, called the run. Pick any two points on a line, draw the little right triangle between them, and divide.

From (1, 1) to (3, 5) the line rises 4 while it runs 2, so its slope is 4 ÷ 2 = 2.Open in grapher →

Going from $(1, 1)$ to $(3, 5)$ , the line rises $4$ and runs $2$ , so the slope is $\dfrac{4}{2} = 2$ . Pick two different points on the same line and you'll get a bigger or smaller triangle, but the ratio is still $2$ . That's what makes a line a line: its slope is the same everywhere.

Definition

Slope

The slope $m$ of the line through $(x_1, y_1)$ and $(x_2, y_2)$ is

m = \frac{\text{rise}}{\text{run}} = \frac{y_2 - y_1}{x_2 - x_1}, \qquad x_1 \ne x_2.

The letter $m$ is the traditional name for slope. The little numbers are subscripts: $x_1$ just means "the $x$ -coordinate of the first point." They are labels, not exponents.

Worked example: Slope from two points

Find the slope of the line through $(-2, 5)$ and $(4, -1)$ .

Call $(-2, 5)$ the first point and $(4, -1)$ the second.

m = \frac{-1 - 5}{4 - (-2)} = \frac{-6}{6} = -1.

The slope is $-1$ : every step of $1$ to the right takes the line $1$ unit down.

Common mistake

Stay consistent. Whichever point's $y$ you write first on top, its $x$ must come first on the bottom. Mixing them, as in $\dfrac{-1 - 5}{-2 - 4}$ , flips the sign and gives $+1$ , which is wrong. Also watch double negatives: $4 - (-2) = 6$ , not $2$ .

It doesn't matter which point you call "first," as long as you're consistent. Using $(4, -1)$ first gives $\dfrac{5 - (-1)}{-2 - 4} = \dfrac{6}{-6} = -1$ , the same answer.

Four kinds of slope

Read a graph from left to right, the way you read a sentence.

Positive slope: the line goes up. As $x$ increases, $y$ increases.
Negative slope: the line goes down. As $x$ increases, $y$ decreases.
Zero slope: the line is horizontal. The rise is $0$ , so $m = \dfrac{0}{\text{run}} = 0$ .
Undefined slope: the line is vertical. The run is $0$ , and you can't divide by zero.

Positive slope (y = 2x + 1), negative slope (y = −x/2 + 3), zero slope (y = −2) and undefined slope (x = 4).Open in grapher →

The bigger the absolute value of the slope, the steeper the line. A slope of $-5$ is steeper than a slope of $2$ , because $|-5| = 5 > 2$ .

Worked example: Horizontal and vertical lines

Find the slope of the line through each pair of points.

$(-3, 4)$ and $(5, 4)$
$(2, -1)$ and $(2, 6)$

Solutions.

$m = \dfrac{4 - 4}{5 - (-3)} = \dfrac{0}{8} = 0$ . The $y$ -values match, so the line is horizontal.
$m = \dfrac{6 - (-1)}{2 - 2} = \dfrac{7}{0}$ , which is undefined. The $x$ -values match, so the line is vertical.

Tip

"Zero slope" and "no slope" sound alike but are different. A horizontal line has a slope, and it equals $0$ . A vertical line's slope is undefined. Saying "undefined" avoids confusion.

Slope as a rate in context

In a real situation, slope has units: the units of $y$ per unit of $x$ . Always say what the slope means.

Worked example: Interpreting slope

At 9:00 a.m. a car's tank holds $14$ gallons. At 1:00 p.m., after steady highway driving, it holds $6$ gallons. Find the rate of change and explain what it means.

Let $x$ be hours after 9:00 and $y$ be gallons. The points are $(0, 14)$ and $(4, 6)$ .

m = \frac{6 - 14}{4 - 0} = \frac{-8}{4} = -2.

The rate of change is $-2$ gallons per hour: the car uses $2$ gallons of fuel every hour. The negative sign says the amount of fuel is decreasing.