Positive Slope

What a positive 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, 2) and (3, 6). The rise is 6 − 2 = 4 and the run is 3 − 1 = 2, so the slope is 4 ÷ 2 = 2. It came out above zero, so the slope is positive and the line rises.

The sign is doing one job and one job only: it tells you the direction. A positive slope means x and y move together. Step to the right and the line goes up with you. Step left and it comes down. The size of the number is a separate matter, covered further down.

Telling a positive slope from a negative one

This is the question people actually arrive with, so here are the three places you will meet it.

	Positive slope	Negative slope
The line	Rises left to right	Falls left to right
The value of m	Greater than 0	Less than 0
x and y	Move the same way	Move opposite ways
Example	y = 2x + 1	y = −2x + 1

• From a graph. Put a finger at the left end of the line and trace right. If your finger goes up, the slope is positive. If it drops, it is negative.
• From an equation. In y = mx + b, look at the number in front of x. In y = 3x − 7 it is 3, so positive. In y = −0.5x + 4 it is −0.5, so negative.
• From two points. Work out the rise over the run. If the answer is above zero the slope is positive.

Why we read left to right, and whether the order matters

Explanations of slope tell you to read the graph from left to right, and it is rare for one to say why. It is worth a minute, because the moment you wonder about it, the whole positive-and-negative business starts to feel shaky.

The reason is in the axes. On the standard coordinate plane, x increases as you move to the right. So reading a line from left to right is not a stylistic choice, it is a way of asking one particular question: as x increases, what does y do? If y rises with it, the slope is positive. If y comes down, the slope is negative.

You will also meet the memory aid that we read a line the way we read words in English, which OpenStax uses and which does help. Just keep it as the reminder rather than the reason. The working reason is the direction of the x-axis.

The reassuring part is that the number itself does not depend on the habit. Take the same two points, (1, 2) and (3, 6), and work through them backwards:

left to right: (6 − 2) ÷ (3 − 1) = 4 ÷ 2 = 2
right to left: (2 − 6) ÷ (1 − 3) = −4 ÷ −2 = 2

Both the top and the bottom changed sign, and the two minus signs cancelled. So it does not matter which point you call the first one. Start wherever you like and you will get the same slope.

Is a slope of −5 steeper than a slope of 2

Yes, and this catches people out because the minus sign makes −5 look like the smaller number. As a number it is. As a slope it is the steeper line.

Slope carries two pieces of information at once, and it helps to read them separately:

• The sign tells you the direction. Plus means rising, minus means falling.
• The size tells you the steepness. Ignore the sign and compare what is left. 5 beats 2, so a slope of −5 is a steeper line than a slope of 2. It just happens to be heading downhill.

Among positive slopes on their own this is easier to see. A slope of 4 climbs faster than a slope of 1, and a slope of 0.2 is a gentle rise that takes five steps sideways to gain one step of height. As the number shrinks towards zero the line flattens out, and at exactly zero it is horizontal.

Where positive slopes turn up

Outside a maths question, a positive slope is any relationship where one thing rises along with another over the stretch you are looking at. Savings growing week by week. A child getting taller through childhood. Distance covered increasing while a car keeps moving. In each case the graph climbs as you move right, because both quantities are going up together.

It shows up in physical objects too. A wheelchair ramp, a staircase and a driveway each have a slope you can put a number to. Whether that number comes out positive depends on which way you measure: walk up the ramp and it rises, walk back down the same ramp and it falls. The steepness has not changed, only the direction you chose, which is the same split as before. If you want the number for a real ramp rather than a line on a graph, the ramp slope calculator works it out from the height and the length.

Frequently asked questions

How do you know if a slope is positive or negative?

Read the line from left to right. If it rises, the slope is positive. If it falls, the slope is negative.

From an equation, look at the number in front of x once y is on its own: y = 3x + 1 is positive, y = −3x + 1 is negative. From two points, work out the rise over the run and check whether the answer lands above or below zero.

What is the formula for a positive slope?

There is no separate formula. Positive slopes use the same one as every other slope: (y₂ − y₁) ÷ (x₂ − x₁).

What makes it positive is the result rather than the method. If the answer comes out above zero, you have a positive slope. The condition is usually written as m greater than 0.

Does it matter which point you subtract first?

No, as long as you are consistent. Working from (1, 2) to (3, 6) gives 4 ÷ 2 = 2. Working from (3, 6) back to (1, 2) gives −4 ÷ −2 = 2. Both the rise and the run flip sign, and the two minus signs cancel.

The mistake to avoid is mixing them: taking the y values one way round and the x values the other. That flips only the top, and the answer comes out with the wrong sign.

What does a positive slope look like on a graph?

A line running uphill from the bottom left towards the top right, like a hill you are walking up rather than down.

The steeper the climb, the larger the number. A gentle rise is a small positive slope such as 0.2, and a sharp climb is a large one such as 6.

Is a slope of −5 bigger than a slope of 2?

It depends what you mean by bigger, which is why the question feels slippery. As a number, −5 is smaller than 2. As a line, it is the steeper of the two.

Read the two parts separately. The sign gives the direction, so −5 is heading downhill. The size, ignoring the sign, gives the steepness, and 5 is steeper than 2.

Can a positive slope be a fraction or a decimal?

Yes. A slope of 1/2 or 0.5 is positive, and so is 3/4 or 0.05. Anything above zero counts, whole number or not.

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

Why does 2x + 3y = 6 have a negative slope when there are no minus signs?

Because in that arrangement the slope is not yet showing as the number in front of x. It is in the equation, but you have to rearrange before you can read it off.

Rearrange it: 3y = −2x + 6, then y = −2/3 x + 2. The slope is −2/3, so the line falls. Whenever an equation is written with x and y on the same side, sort it into y = mx + b before reading anything from it.

What does a positive slope mean in real life?

That two things are going up together. Savings rising week by week, height increasing with age, distance growing as time passes.

Slope is a rate of change, so a positive slope is a rate of increase. The larger the slope, the faster the increase.