When you hear the word "slope," you might picture a mountain path, a ski run, or a line on a graph going up or down. A positive slope is the uphill version. It starts low and rises steadily as you move to the right.
Whether you’re learning about a positive slope for the first time, preparing for a test, or looking to get ahead in your math class, this guide is for you.
Mathnasium tutors walk you through what a positive slope is with step-by-step instructions on how to calculate it, with worked examples, a quiz, and answers to the most common questions students have about positive slopes.
A positive slope means a line rises as it moves from left to right on a graph.
As x-values increase, y-values also increase on a line with a positive slope.
We calculate the slope using the equation m = \(\Large\frac{rise}{run}\) = \(\Large\frac{Δy}{Δx}\).
For a positive slope, both the rise (Δy) and the run (Δx) are always positive.
A line with a positive slope forms an acute angle with the x-axis (between 0° and 90°).
The steeper the line, the larger the value of the positive slope.
There are four types of slope: positive, negative, zero, and undefined.
In math, the slope of a line tells us how steep a line is and how it moves across a graph. More specifically, 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 call this rise over run:
m = \(\Large\frac{rise}{run}\) = \(\Large\frac{Δy}{Δx}\)
Let's see what each part is telling us:
m: the symbol we use to represent the slope in equations
rise: the change (Δy) in y-values, or how much the line goes up or down
run: the change (Δx) in x-values, or how much the line moves left or right
In math, we use the Greek letter Δ to denote a change in a variable.
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In math, we usually work with four types of slopes:
Positive Slope: A line with a positive slope rises as it moves from left to right on a coordinate plane. Both the rise (Δy>0) and the run (Δx>0) are positive, which means the line moves upward. For example, it’s like walking uphill.
Negative Slope: A line with a negative slope falls as it moves from left to right on a coordinate plane. The rise (Δy<0) is negative because the line drops, while the run (Δx>0) is still positive. For this one, we are going downhill.
Zero Slope: A line with zero slope is flat, like a horizontal road. There's no rise (y = 0), so the slope equals zero.
Undefined Slope: A line with an undefined slope is vertical, like a wall. There's no run (x = 0), which makes the slope undefined because we can't divide by zero.
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A positive slope describes a line on a graph that moves upward as it goes from left to right. This happens when the line's y-values increase while the x-values also increase.
Simply speaking, for every step forward along the x-axis, the line rises higher on the y-axis.
A line with a positive slope forms an acute angle with the x-axis. This means the angle it forms relative to the x-axis is less than 90° (denoted as 0°<θ<90°). The steeper the positive slope, the closer this angle gets to 90°.
To determine whether a slope is positive or negative, we calculate the rise over run:
m = \(\Large\frac{rise}{run}\) = \(\Large\frac{Δy}{Δx}\)
For a positive slope:
The rise (Δy) is positive because the line moves upward.
The run (Δx) is positive because the line moves to the right.
This creates a positive ratio, \(\Large\frac{2}{3}\) for example, which means the line rises 2 units for every 3 units it moves forward.
To see exactly what a positive slope looks like, let’s take a look at an image below.
What is this image telling us?
The line forms an acute angle (0°<θ<90°) with the x-axis.
The line passes through two specific points: (2, 1) at the lower point and (4, 4) at the higher point. We notice that these are the points where the line crosses the grid.
Since we see that on the x-axis, the line moves two units to the right, so the run or Δx = 2, and it moves three units up along the y-axis, so the rise or Δy = 3.
We conclude that with a positive rise and a positive run, the slope of our line is positive.
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We calculate the slope by dividing the rise (the change along the y-axis) by the run (the change along the x-axis):
m = \(\Large\frac{rise}{run}\) = \(\Large\frac{Δy}{Δx}\)
We can represent these changes using specific points (x1, y1) and (x2, y2):
Δy =y2 - y1: the vertical change between the two points.
Δx =x2 - x1: the horizontal change between the same points.
If we want to calculate the slope of a line, we simply look at the graph, read the coordinates along the x and y axes, and determine the change like so:
m = \(\Large\frac{Δy}{Δx} = \Large\frac{y2 - y1}{x2 - x1}\)
What happens if we don't have a graph in front of us? As long as we know the coordinates, we can still calculate the slope.
We can tell whether the slope is positive or negative based on the result.
In simple terms, if the fraction \(\Large\frac{Δy}{Δx}\) is positive (with results like \(\Large\frac{2}{3}\), 2, or 1.5), that means the slope is positive.
Let's put this to the test.
We have two points on a line:
(x1, y1) = (1, 2)
(x2, y2) = (4, 8)
First, we use the slope equation:
\(m = \Large\frac{y2 - y1}{x2 - x1}\)
Then, we substitute the values:
\(m = \Large\frac{8 - 2}{4 - 1} = \Large\frac{6}{3}\)
Then, we can divide 6 by 3, and the result of our fraction is a whole number:
m = 2
Since we've calculated a positive 2, the slope is positive, and the line rises as it moves from left to right.
Let’s go through a few solved examples of positive slope together!
Let’s start with a visual example. By looking at the graph, let’s determine the changes in the positive slope step by step.
On the graph, our line passes through two specific points:
(x1, y1) = (2, 1)
(x2, y2) = (4, 5)
First, we can visually trace how far the line moves to the right along the x-axis. It goes from 2 to 4. That's 2 units, so our run is Δx = 2.
Then, we can visually trace how far the line moves up along the y-axis. It goes from 1 to 5. That's 4 units, so our rise is Δy = 4.
Since \(m = \Large\frac{Δy}{Δx}\), we now substitute the values:
\(m = \Large\frac{4}{2} = 2\)
The slope of the line is 2.
Now, without a graph, let's calculate the slope of a line and determine if it's positive. Our line passes through the points:
(x1, y1) = (0, 1)
(x2, y2) = (3, 4)
Since we aren't looking at a graph, we'll use the coordinate-based slope equation:
\(m = \Large\frac{y2 - y1}{x2 - x1} = \Large\frac{4 - 1}{3-0} = \Large\frac{3}{3} = 1\)
Our result is positive, and that means the slope is positive as well.
Ready to practice what we’ve covered? Give the positive slope quiz a try on your own and check your answers at the bottom of the guide.
1. Which of these statements describes a positive slope?
The line falls as it moves to the right on a coordinate plane.
The line rises as it moves to the right on a coordinate plane.
The line stays flat as it moves to the right.
The line is vertical.
2. Which of these is an example of a positive slope in real life?
A downhill road.
A flat sidewalk.
A staircase going upward.
A vertical pole.
3. What is the angle formed by a positive slope with the x-axis?
Less than 90°
Exactly 90°
Greater than 90°
Between 90° and 180°
4. The line passes through the points (1, 2) and (4, 5). What is the slope of this line?
-1
1
\(\Large\frac{2}{3}\)
3
Here are some of the most common questions we hear from our students about a positive slope, and the answers to help clear up any confusion.
Yes. A larger positive number (e.g. 5) represents a steeper upward slope, while a smaller positive number (e.g. \(\Large\frac{1}{2}\)) represents a gentler upward slope.
Yes. By using the slope formula \(m = \Large\frac{Δy}{Δx}\) with any two points from the line, you can calculate the slope even without the entire graph.
If the rise and run are equal, the slope will always be 1. This means the line rises at the same rate as it moves to the right, and always forms a 45° angle with the x-axis.
For instance, let’s take a look at the graphic below.
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If you worked through the quiz, here are the answers:
b) The line rises as it moves to the right on a coordinate plane.
c) A staircase going upward.
a) Less than 90°
b) 1
How did you do?
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