What is a Negative Slope? Definition, Graphs, and Examples

A negative slope defines a line slanting downward from left to right on a graph. In coordinate geometry, this means that as x-values increase, y-values decrease at a specific rate given by a linear equation.

The concept of a negative slope 🇺🇸 is fundamental in math, particularly in algebra and graphing, as it is used in linear equations, coordinate geometry, and calculus. But what is the definition of negative slope, and what does a negative slope look like? Thanks to this article, you will discover what a negative slope is and have a chance to look at negative slope graphs and examples.

Yes, a slope can absolutely be negative. Let's take a look at the slope-intercept form 🇺🇸 of a linear equation:

$y = mx + b$

As its name suggests, an equation of this form indicates the slope, $m$, and the y-intercept, $b$, of the linear function. For example, for the equation $y = 5x + 4$, the slope is equal to $5$, while the y-intercept is $(0, 4)$.

What is a negative slope?

In graphic terms, a negative slope is a line that moves downward from left to right. But what does that mean for our linear equation? Whenever there is a decreasing linear relationship, $m$ will always be negative.

Negative slope graph: What does a negative slope look like?

Take a look at this graph:

This is a graph of the linear function $y = -2x + 1$; therefore, its slope is equal to $-2$, and as you can see yourself, the function intercepts the y-axis at $(0, 1)$, just like the slope-intercept form suggests.

Here are some key characteristics that will help you identify a negative slope without fault:

• It starts high on the left and goes downward toward the right.
• The slope gets steeper as $m$ decreases.
• Conversely, as $m$ gets closer to $0$, the line becomes flatter.

Let's modify the slope a little bit to see how $m$ shapes the slope:

This is a graph representing the function $y = -\frac{1}{5}x + 1$. This time, $m$ is much closer to $0$ compared to $-2$; therefore, the slope is less steep.

You can find a negative slope in two ways: either identify it from a line equation in the slope-intercept form, or calculate it using the coordinates of two points 🇺🇸 belonging to the line.

Identifying a negative slope from a line equation

Do you remember the slope-intercept form of a line equation? Here it is as a reminder:

$y = mx + b$

Whenever you find an equation of this form, you can be sure that $m$ represents the slope of the line. Let's take a look at an example:

$y = -4x - 8$

Can you identify the negative slope in this case? Yes, the slope is equal to $-4$. But let's see what happens if the equation has the following form:

$2x + 3y = 6$

This is the standard form of a linear equation, $x$ and $y$ are on the same side, and there is a $3$ before the $y$. To find the slope of a line equation in its standard form, proceed as follows:

1. Rearrange the equation such that the term containing $y$ is alone on the left side:

   $3y = -2x + 6$

2. Perform the necessary operations to eliminate any numbers surrounding $y$. In this case, we need to divide both sides of the equation by $3$:

   $\large\frac{3y}{3} = \frac{-2x + 6}{3}$

   $y = -\frac{2}{3}x + 2$

   Now the equation is in slope-intercept form, and we can easily spot the slope.

3. Identify $m$. In this particular equation, $m = -\frac{2}{3}$.

And that's it! All you need to remember is that if you have a line equation and want to identify the negative slope, you need to rearrange the equation — if necessary — to make sure that $y$ is alone on its left side, then identify $m$. If $m < 0$, the slope is negative.

Calculating a negative slope using the coordinates of two points

If you are not given the equation but know two points on the line, you can still find the slope using the rise over run formula 🇺🇸.

We can use the rise over run formula to calculate our slope given two points, $A(x_1, y_1)$ and $B(x_2, y_2)$:

$\large m = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1}$

Let's see an example. Consider two points, $A(7, 10)$ and $B(9, 4)$. How would you calculate the slope using the coordinates $A$ and $B$?

1. Subtract the y-coordinate of $B$ from that of $A$:

   $y_2 - y_1 = 4 - 10 = -6$
   ​

2. Subtract the x-coordinate of $B$ from that of $A$:

   $x_2 - x_1 = 9 - 7 = 2$
   ​

3. Divide the result of step 1 by that of step 2 to put the formula together:

   $m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-6}{2} = -3$
   ​

Given that $m < 0$, the slope is negative, which we can also see from the graph:

The line equation passing through $A(7, 10)$ and $B(9, 4)$ is $y = -3x + 31$.

A negative slope means that a line decreases as it moves from left to right, showing that y-values drop as x-values increase. This concept is essential in algebra and graphing. You can find the slope of a line by identifying it in a slope-intercept form of the equation, or using two points belonging to the line.

In this article, you explored some examples of negative slopes and graphs to help you recognize and work with negative slopes without any issues.