What Is a Negative Slope? A Beginner’s Guide

Jan 30, 2025 | 4S Ranch
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When you hear the word "slope," you might think of mountain slopes, a path, or a line going up or down. A negative slope is like skiing or going downhill—it starts at a higher point and steadily drops as you move forward.

This guide takes a closer look at the math behind negative slopes. Whether you’re learning about them for the first time, preparing for a test, or looking to get ahead in your math class, stick around.

Read on for simple definitions, easy-to-follow instructions, worked-out examples, a fun quiz, and answers to common questions to help you master negative slopes. 

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What Is a Slope in Math?

In math, the slope of a line tells us how steep a line is and how it moves across a graph. 

It’s a way to measure how much the line goes up or down (on the y-axis) compared to how far it moves sideways (on the x-axis).

We determine the slope of a line by calculating its rise over run like so:

m = \(\Large\frac{rise}{run}\) = \(\Large\frac{Δy}{Δx}\)

Let’s break this down:

  • m: This is the symbol we use to represent the slope in equations.

  • rise = Δy:  The "change in y," or how much the y-values go up or down.

  • run = Δx: The "change in x," or how much the x-values move left or right.

Types of Slopes

When we look at a line on a graph, its slope can fall into one of four categories. 

Each type of slope tells us something different about how the line behaves:

  1. Positive Slope: A line with a positive slope rises as it moves to the right on a coordinate plane. It shows that as the x-values increase, the y-values also increase. Think of it as going uphill!

  2. Negative Slope: A line with a negative slope falls as it moves to the right on a coordinate plane. It shows that as the x-values increase, the y-values decrease. Think of it as going downhill.

  3. Zero Slope: A flat, horizontal line has a zero slope. It means there’s no change in y-values as the x-values increase. The line stays level, like a calm and steady road.

  4. Undefined Slope: A vertical line has an undefined slope. This is because the x-values don’t change, while the y-values might change infinitely. It’s like climbing straight up a wall! So, think of Spiderman.

Illustration of types of slopes.


What Is a Negative Slope?

A negative slope describes a line on a graph that moves downward as it goes from left to right. 

This happens when the line's y-values decrease while the x-values increase. In simple words, for every step forward (to the right) along the x-axis, the line drops lower on the y-axis.

Illustration of types of slopes.

A line with a negative slope forms an obtuse angle with the x-axis. This means the angle is greater than 90° but less than 180° (denoted as 90º < θ < 180º). 

In addition to visually assessing a line on a graph, we can determine whether our slope is negative or positive by calculating the line’s rise over run:

m = \(\Large\frac{rise}{run}\) = \(\Large\frac{Δy}{Δx}\)

For a negative slope:

  • The rise (Δy) is always negative because the line drops.

  • The run (Δx) is always positive because the line moves to the right.

This creates a negative ratio, such as \(\Large\frac{-2}{3}\), which means the line drops 2 units for every 3 units it moves forward.

To see what this looks like, let’s visualize a negative slope on a graph.

Example of a negative slope.


How Do We Calculate a Negative Slope?

We can figure out the slope of a line by dividing the rise (the changes along the y-axis) with the run (the changes along the x-axis):

m = \(\Large\frac{rise}{run}\) = \(\Large\frac{Δy}{Δx}\)

What does “the change along the axis” mean? – you may ask. 

When you see the Greek letter Δ in math, it usually means that we are looking to determine or denote a change in a variable.

The change in y (y) is how much the y-values differ between two points on the line. 

Similarly, the change in x (x) is how much the x-values differ between two points.

We can represent these changes using specific points (x1, y1) and (x2, y2), where:

  • y = y2 - y1:  the vertical change between the two points.

  • x = x2 - x1: the horizontal change between the same points.

So, if we wanted to calculate the slope of a line, all we have to do is look at the graph, read the coordinates along the x and y axes, and figure out the change like so:

m = \(\Large\frac{Δy}{Δx}\) = \(\Large\frac{y2-y1}{x2-x1}\)

What happens if we don’t have a graph? 

As long as we know the coordinates, we can calculate the slope. And, guess what? You can tell whether the slope is positive or negative based on the result.

For a negative slope:

  • y2 - y1 (the rise), which represents the numerator in our fraction, is always negative because the line drops.

  • x2 - x1 (the run), which represents the denominator in our fraction, is always positive because the line moves to the right.

Since our numerator is negative and our denominator positive, the result of \(\Large\frac{y2-y1}{x2-x1}\) is always negative (for example, \(\Large\frac{-2}{3}\) or -1) for negative slopes.

Let’s test this!

Take a look at the graph of a line passing through the points:

(x1, y1) = (0, 3)

(x2, y2) = (3, 1)

Solving for negative slopes.

With a graph like this in front of us and our coordinates marked, we don't even need to calculate—we can simply "read" our slope.

  • We start at (0, 3), the higher point, and count how far down the line drops to reach (3, 1). This is two units. This will be our rise (Δy = -2).

  • Next, we count how far the line moves to the right. This would be 3 units. This will be our run (Δx = 3).

Solving for negative slopes 2.

Visually, we can see that the slope is: = \(\Large\frac{-2}{3}\).

Now, let’s confirm our result by using the slope formula:

m = \(\Large\frac{y2-y1}{x2-x1}\)

Substitute the points (x1, y1) = (0, 3) and (x2, y2) = (3, 1):

= \(\Large\frac{1-3}{3-0}\) = \(\Large\frac{-2}{3}\) 


Solved Examples of Negative Slope

Now that we know what negative slopes are and how to calculate them, let’s see some examples!


Example 1

Let’s determine if the slope is negative if the line passes through the points (2, 5) and (6, 1), left to right.

  • (x1, y1) = (2, 5)

  • (x2, y2) = (6, 1)


We will determine the type of slope by figuring out the changes along the y and x-axis:
= \(\Large\frac{y2-y1}{x2-x1}\)

= \(\Large\frac{1-5}{6-2}\)

= \(\Large\frac{-4}{4}\)


We can simplify the fraction \(\Large\frac{-4}{4}\) by dividing the numerator by the denominator.

\(\Large\frac{-4}{4}\) = -1

= -1

The slope is negative (-1), meaning the line moves downward as it travels from left to right.


Example 2

Let’s determine if the slope is negative if the line passes through the points (1, 3) and (4, 9), left to right.

  • (x1, y1) = (1, 3)

  • (x2, y2) = (4, 9)


We will determine the type of slope by figuring out the changes along the y and x-axis:

= \(\Large\frac{y2-y1}{x2-x1}\) 

= \(\Large\frac{9-3}{4-1}\)

= \(\Large\frac{6}{3}\)


We simplify the fraction \(\Large\frac{6}{3}\) by dividing the numerator by the denominator:

6 ÷ 3 = 2

= 2

We calculated a positive number (2), which tells us the slope is not negative.


Quiz: Test What You’ve Learned About Negative Slope


Ready to put your knowledge to the test? 

Give these a try and check your answers at the bottom of the guide!

  1. Which of the following is NOT one of the four types of slope?

a) Positive slope

b) Curved slope

c) Negative slope

d) Undefined slope


  1. How can you tell if a slope is negative?

a) The rise and run are both positive.

b) The rise is positive, and the run is negative.

c) The rise is negative, and the run is positive.

d) The slope equals zero.


  1. What is the angle formed by a negative slope with the x-axis?

a) Less than 90º

b) 90º

c) Between 90º and 180º

d) Greater than 180º


  1. The line passes through the points (2, 5) and (6, 1). What is the slope of this line?

a) -1

b) 1

c) \(\Large\frac{-4}{3}\)

d) \(\Large\frac{3}{4}\)


FAQs About Negative Slope

Learning about negative slopes in math often raises interesting questions for students. 

Here are some of the most common questions we hear at Mathnasium of 4S Ranch and the answers to help clear up any confusion!


1) Can a slope be “more negative” or “less negative”?

Yes!

A slope with a larger negative value (e.g. -5) is steeper than a slope with a smaller negative value (e.g. \(\Large\frac{-1}{2}\)). 

Both slopes move downward, but the larger the absolute value, the steeper the decline.


2) What happens if the rise and run are both negative?

If both the rise (y) and run (x) are negative, the negatives cancel out, resulting in a positive slope. 

If a line moves downward and to the left, it has a positive slope–it’s just a matter of perspective! 


3) Can a horizontal line ever have a negative slope?

No, a horizontal line always has a slope of zero because there’s no rise (y = 0) regardless of the run.


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Pssst! Check Your Answers Here


Q1 - Answer: b) Curved slop

Q2 - Answer: c) The rise is negative, and the run is positive.

Q3 - Answer: c) Between 90º and 180º

Q4 - Answer: a) −1