Negative Slope

What a negative slope means on a line

Slope measures how much a line climbs for each step you take sideways. Take two points, subtract to get the rise and the run, and divide.

slope = (y₂ − y₁) ÷ (x₂ − x₁)

Take (1, 5) and (4, 2). The rise is 2 − 5 = −3 and the run is 4 − 1 = 3, so the slope is −3 ÷ 3 = −1. The answer came out below zero, so the slope is negative and the line falls.

Notice that the word “rise” is doing something odd here. The rise came out at −3, which is a fall of 3 rather than a climb. That is normal and it is worth getting comfortable with: rise is just the change in y, and a change can go either way. A negative rise is the line dropping.

Where the minus sign actually lives

Most of the trouble on this topic comes from minus signs turning up where they do not belong, or hiding where you expect them. Three cases are worth knowing, because a question can be built on any of them.

The second case is the reverse. In y = 5 − 2x the minus is easy to miss because it is tucked in the middle. Rewrite it as y = −2x + 5 and the slope is plainly −2. Whenever the terms are out of the usual order, put them back into y = mx + b before you read anything off.

The third is the one that costs marks. Two minus signs in the slope formula cancel, and a negative divided by a negative is positive. Work out the slope between (4, 2) and (1, 5) in that order: the rise is 5 − 2 = 3 and the run is 1 − 4 = −3, giving 3 ÷ −3 = −1. Still negative, which is correct. But if you subtract the y values one way round and the x values the other, you will get +1 and conclude the line rises. Keep both subtractions in the same order and the sign takes care of itself.

Which is steeper, a slope of −2 or −5

−5 is the steeper line, even though it is the smaller number. The confusion usually starts with the word people reach for, which is not “steeper” but “bigger”. Ask which of −2 and −5 is bigger and the question has two defensible answers, which is why it feels like a trap.

The fix is to stop reading the slope as one number and start reading it as two pieces of information.

• The sign is the direction. Below zero means falling. That part is settled the moment you see the minus, and nothing else about the number changes it.
• The size is the steepness. Cover the minus sign and compare what is left. 5 is bigger than 2, so −5 drops more sharply than −2.

A slope of −0.1 is a gentle downward drift, close to flat. A slope of −20 is close to vertical. Both are negative, and the second has two hundred times the size of the first. The same rule works when comparing across the sign, so −5 is steeper than a positive slope of 2, and one goes down while the other goes up.

Three ways to spot a negative slope

• From a graph. Start at the left end of the line and trace right. If your finger drops, the slope is negative.
• From an equation. Once y is on its own, look at the number in front of x. If it carries a minus sign, the slope is negative. In y = −0.5x + 4 it is −0.5.
• From two points. As x grows, y shrinks. In (1, 5) and (4, 2) the x values go up while the y values come down, which is the pattern every negative slope makes.

One case sits outside all three. A vertical line does not fall or rise as x increases, because x never increases: every point sits at the same x value. Its slope is not negative, and not zero either. It is undefined, which means no number belongs there at all.

Where negative slopes turn up

Anywhere one thing shrinks while another grows. A phone battery draining through the afternoon. A car losing value each year it is owned. A tank emptying while the tap runs. Plot any of them against time and the line heads downwards, because the second quantity is falling as the first climbs.

Economics leans on this heavily, which is why the phrase turns up in coursework that has nothing to do with algebra. A demand curve slopes downwards because, holding other things constant, higher prices generally go with a lower quantity demanded, and describing that curve as having a negative slope is just saying the two move in opposite directions. One detail catches people out when they try to line the two subjects up, and it is covered in the questions below.

The word also has a life in building work, where it means something more specific and more expensive. See the last question below if that is what brought you here.

Frequently asked questions

What is the slope of y = −4?

Zero. Despite the minus sign, y = −4 is a horizontal line and horizontal lines have a slope of 0.

The −4 says where the line sits, four units below the x-axis, not how steeply it climbs. Slope is attached to the x term, and this equation has no x in it, so there is nothing to climb. Written in full it is y = 0x − 4.

Does a negative slope mean the line is decreasing?

Yes. For a straight line the two phrases describe the same thing: as x increases, y decreases.

The wording differs because “decreasing” is usually used of the quantity and “negative slope” of the line drawn from it. On a curve the idea still holds but only piece by piece, since a curve can fall over one stretch and rise over another.

How do I know if my slope is negative or positive?

Read the line from left to right. Falling means negative, rising means positive. In an equation, check the sign of the number in front of x once y is on its own.

From two points, see whether y goes up or down as x goes up. If they move in opposite directions the slope is negative.

Which is bigger, a slope of −2 or a slope of −5?

As numbers, −2 is the greater of the two, because it sits further to the right on a number line. As lines on a graph, −5 is the steeper, because it has the greater size once the sign is set aside.

So answer −2 if the question asks which slope is greater, and −5 if it asks which line is steeper. If it only says “bigger”, say which reading you have taken, because the word carries both. In your own working, write “steeper” and the ambiguity never comes up.

What does a negative slope look like on a graph?

A line running downhill from the top left towards the bottom right, like a slide.

The steeper the drop, the larger the number after you ignore the minus sign. A slope of −0.2 is a gentle slide downwards and a slope of −6 is a sharp one.

Can a negative slope be a fraction?

Yes. −1/2, −2/3 and −0.05 are all negative slopes. Anything below zero counts, whole number or not.

A slope of −2/3 means the line drops 2 for every 3 you move across. Fractions are common because rise over run rarely divides neatly, and the fraction is usually a better answer than a rounded decimal.

What does a negative slope mean in economics?

That the two quantities on the axes move in opposite directions. The usual example is a demand curve, where a higher price generally goes with a lower quantity demanded, other things being held constant, so the curve heads downwards.

There is a catch worth knowing about, because it is the thing that makes economics graphs feel wrong to anyone who learned slope in a maths class. The axes are the other way round. Quantity is drawn along the bottom and price up the side, even though quantity is the thing that depends on price. OpenStax says so plainly in its economics textbook, calling it an exception to the normal rule in mathematics that the independent variable goes on the horizontal axis.

So the slope you read off a demand curve is the change in price for each unit of quantity, which is upside down compared with the more natural question of how much quantity shifts for each pound on the price. The direction is unaffected, and the curve is negative either way. The number is not the one a maths student expects, and that is usually what the confusion turns out to be.

What does “negative slope” mean for a drain or a sewer pipe?

Something quite different from the algebra. In drainage work the phrase is generally used for reverse grade: a length of pipe that climbs in the direction the water is meant to travel, so waste backs up instead of running away. Three conditions get discussed together and are worth keeping apart. A flat run has zero grade. A reverse run rises against the flow. A sag, or belly, is a dip that can contain both, falling into the low point and climbing out of it.

One practical caution, since the maths and the trade meet here. The sign a survey or a camera report gives depends on which end the tool treats as the start, so a reading of negative slope is worth checking against the intended direction of flow before anyone digs. If you want the number for a run of pipe, the pipe slope calculator works it out from the fall and the length.