Positive Slope vs. Negative Slope: How to Tell the Difference

The difference between a positive and a negative slope is simple: a positive slope rises from left to right, while a negative slope goes down. However, visual cues aren't the only way to tell the difference between a positive and a negative slope. In this article, we'll learn how to tell if a slope is positive or negative, which will help you differentiate between the two without any issues, both visually and algebraically.

The easiest way to distinguish between a positive and a negative slope is by looking at a graph — if the slope is positive, the line will slant upward, while if it's negative, it will move downward from left to right. Nevertheless, mathematical ways exist to compare a positive slope vs. a negative slope.

How to tell if a slope is positive or negative

Here is a quick overview comparing positive slope vs negative slope:

Each type of slope creates a distinctive shape on a graph and can be calculated and compared algebraically, using equations or points. So, how can you calculate m to tell whether the slope is positive or negative? Let's dive into it!

Identifying m in a line equation

Linear equations are often written in the slope-intercept form 🇺🇸:

$y = mx + b$

In this type of equation, the coefficient $m$ represents the slope. If $m>0$, the slope is positive, which means that there is a positive correlation between $x$ and $y$: y-values increase as x-values increase. On the other hand, if $m<0$, the slope is negative, which represents a negative correlation between $x$ and $y$. Therefore, y-values will decrease as x-values increase.

Let's look at some examples. Consider the equation $y = -\frac{3}{7}x + 2$.

$m = -\frac{3}{7}$, which is less than 0; therefore, the slope is negative, which we can see on the graph of the equation:

Now look at this equation: $y = 6x - \frac{1}{3}$. Is the slope positive or negative?

That's right! Since $m > 0$, the slope of this line is positive.

As you can see, this is confirmed by the graph, since the line goes upward from left to right.

Now let's take a different equation.

$3x + \frac{1}{2}y = 6$

This is the standard form 🇺🇸 of a linear equation. Here, $y$ is no longer isolated on the left side, which makes it impossible to determine the slope. But don't worry, all you have to do in this case is rearrange the equation such that $y$ is alone on the left side:

$\frac{1}{2}y = -3x + 6$

$\frac{1}{2}y \times 2 = (-3x + 6)\times 2$

$y = -6x + 12$

Now we know that the slope is equal to $-6$, and since $m<0$, the slope is definitely negative.

Calculating and comparing m using two points

When you don't know the equation of a line, check whether you can identify the coordinates of two points belonging to that line, $A(x_1, y_1)$ and $B(x_2, y_2)$. Given these two points, you can use the rise over run formula 🇺🇸 to find $m$ and recognize whether your slope is positive or negative. Here is the formula you have to use:

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

For example, take the points $A(6,12)$ and $B(-2,3)$. Is the slope positive or negative? Let's calculate it:

$\large m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{3 - 12}{-2-6} = \frac{-9}{-8} = \frac{9}{8}$

$\frac{9}{8} > 0$, therefore, the slope is positive.

Now you know everything you need to compare a positive slope vs. a negative slope, whether you are looking at a graph, two distinct points, or a linear equation. Understanding the differences is the key to accurately interpreting any linear relationship.