Gradient is another word for «slope». The higher the gradient of a graph at a point, the steeper the line is at that point. A negative gradient means that the line slopes downwards.
The video below is a tutorial on Gradients.
Finding the gradient of a straight-line graph
In this graph, the gradient = (change in y-coordinate)/(change in x-coordinate) = (8-6)/(10-6) = 2/4 = 1/2
Finding the gradient of a curve
To find the gradient of a curve, you must draw an accurate sketch of the curve. At the point where you need to know the gradient, draw a tangent to the curve. A tangent is a straight line which touches the curve at one point only. You then find the gradient of this tangent.
Find the gradient of the curve y = x² at the point (3, 9).
Note: this method only gives an approximate answer. The better your graph is, the closer your answer will be to the correct answer. If your graph is perfect, you should get an answer of 6 for the above question.
Two lines are parallel if they have the same gradent.
The lines y = 2x + 1 and y = 2x + 3 are parallel, because both have a gradient of 2.
Perpendicular Lines (HIGHER TIER)
Two lines are perpendicular if one is at right angles to another- in other words, if the two lines cross and the angle between the lines is 90 degrees.
Gradient (Slope) of a Straight Line
The Gradient (also called Slope) of a straight line shows how steep a straight line is.
Calculate
To calculate the Gradient:
Gradient = Change in YChange in X
Have a play (drag the points):
Examples:
The Gradient = 33 = 1
So the Gradient is equal to 1
The Gradient = 42 = 2
The line is steeper, and so the Gradient is larger.
The Gradient = 35 = 0.6
The line is less steep, and so the Gradient is smaller.
Positive or Negative?
Going from left-to-right, the cyclist has to Push on a Positive Slope:
When measuring the line:
Gradient = −42 = −2
That line goes down as you move along, so it has a negative Gradient.
Straight Across
Gradient = 05 = 0
A line that goes straight across (Horizontal) has a Gradient of zero.
Straight Up and Down
Gradient = 30 = undefined
Rise and Run
Sometimes the horizontal change is called «run», and the vertical change is called «rise» or «fall»:
They are just different words, none of the calculations change.
Gradient Calculator
Welcome to the gradient calculator, where you’ll have the opportunity to learn how to calculate the gradient of a line going through two points. «What is gradient?» you may ask. Well, have you ever looked at a mountain and said to yourself, «Wow, that mountain is quite steep, but not as steep as the one next to it!«? And if that kind of question has left you wondering how their steepness compares, you’ve come to the right place!
If you want to find the gradient of a non-linear function, we recommend checking the average rate of change calculator.
What is gradient?
Before we take a look at the gradient definition, let’s get back to our mountain scene, and the absolutely crucial question of steepness.
Let’s say you’re skiing down a slope when The Big Question hits you. You stop and think about it before going any further. As we’ve mentioned above, all you need is two points to find the gradient, so why not be a little self-centered and choose yourself as the. well, center, that is, the point (x₁,y₁) = (0,0) on the plane.
Tell the tree or the skier to stand still while you use your handy ruler (that you always carry around with you, of course) to count how much higher/lower they are from you (that will be y₂ ) and how far they are from you (that will be x₂ ). Remember to count the distance between you two horizontally, not parallel to the slope. And there you have it! The ratio of y₂ / x₂ is your gradient, or the steepness of the mountain at that point.
For sticking around while you perform your quick experiment, go and buy that skier some hot chocolate, or give the tree a hug. They deserve as much.
Gradient definition
An informal definition of the gradient (also known as the slope) is as follows: it is a mathematical way of measuring how fast a line rises or falls. Think of it as a number you assign to a hill, a road, a path, etc., that tells you how much effort you have to put to cycle it. If you’re going uphill, you must struggle to reach the peak, so the energy needed (i.e., the gradient) is large. If you’re going downhill, you don’t even have to pedal to pick up speed, so the effort is, in fact, negative. And if you’re on flat ground, it neither helps nor makes it harder, so it is neutral, or has a gradient of zero.
And what if you’re facing a vertical slope? Well, it’s not always clear if you want to fall down it (which is effortless) or go scrambling up it. Therefore, in this case the gradient is undefined.
How to calculate gradient?
To calculate the gradient, we will find two points. We will denote these points with the cartesian coordinates (x₁,y₁) and (x₂,y₂) respectively. This is also the notation used in the calculator. Note that we used the same symbols in the real-life example. We want to see how they relate to each other, that is, what is the rise over run ratio between them. It is described by the gradient formula:
gradient = rise / run
Gradient formula: example of application
Now that we know the gradient definition, it’s time to see the gradient calculator in action and go through how to use it together, step by step:
Common misconceptions and mistakes
You may ask yourself, «Hold on, I think I’ve seen this somewhere else. Doesn’t something similar happen when you count the slope, or the rise over run?» You’re absolutely right. All three of these concepts: gradient, slope, and rise over run describe the same thing, and don’t you worry, there is no difference between them.
You may also wonder how steep is steep; that is, what does the 2 in the above example tell us. Is it a lot, or is it not? Is the pretty skier going to be impressed by this number? Well, it’s all a matter of perspective, and some may say one thing, while others will say the opposite. As a point of reference, you should remember that having a line parallel to the horizon is considered neutral here, as the gradient equals zero. When it rises (or falls), it becomes more and more like a line perpendicular to the horizon, where the slope goes to infinity when it rises (or minus infinity when it is falling).
Gradient Definition
In this mini-lesson, we shall explore the world of the gradient, by finding answers to questions like what is a gradient, what is a directional derivative, and understanding the properties of gradients with examples.
Let us understand how the gradient of line changes with a change in inclination of the line.
In this visualizer, move point B, and observe the change in the gradient of the line.
On moving the point B in the above simulation, the following three changes can be observed.
(a) coordinates of the point B (b) the angle of the line (c) the slope of the line
Lesson Plan
What Is the Definition of Gradient?
The gradient is the inclination of a line. The gradient is often referred to as the slope (m) of the line. The gradient or slope of a line inclined at an angle \(\theta \) is equal to the tangent of the angle \(\theta \).
The gradient can be calculated geometrically for any two points \((x_1, y_1)\), \((x_2, y_2)\) on a line.
Further, the gradient is a quantity, which helps to understand the variation in one quantity, with respect to another quantity.
For a function f(x) the gradient is calculated from its first derivative, \(\frac.f(x) \).
Summarizing the above sentences, we have:
Properties of Gradient
The following properties of a gradient help to understand the orientation of the line.
What Is a Directional Derivative?
For a line drawn in an n-dimensional space, the gradient of the line with reference to a specific dimension is called its directional derivative.
The concept of the partial derivative is helpful to find the directional derivative. And it is represented as \(\frac<\delta y> <\delta x>\)
Let us consider an equation y = 5x + 4z + 3xz + 11
This represents the equation of a line in a 3-dimensional array.
Here the partial derivative with reference to x gives the directional derivative in the direction of x-axis. In this expression, z is treated as a constant.
\(\therefore \) 5 + 3z is the directional derivative of the equation of the line with respect to the x-axis.
The equation y = mx + c is referred as slope-intercept form. Here, «m» is the slope, and «c» is the y-intercept of the line.
Solved Examples
Suzane is trying to climb up a ladder, which is inclined at an angle of \(60^0 \). What is the gradient of the ladder?
Solution
It is given that the inclination is \(60^0 \).
Gradient of the ladder is its slope (m).
\[\begin m &= tan\theta \\ m &=tan60^0 \\ m &=\sqrt3 \end \]
\(\therefore \) The gradient of the ladder is \(\sqrt 3 \)
Example 2
Albert marks two points (4, 3) and (6, 7) on a graph paper and draws a line passing through these points. Find the gradient of the line.
Solution
The given points are \((x_1, y_1) \) = (4, 3) and \((x_2, y_2) \) = (6, 7)
The gradient is the slope(m) of the line joining these points.
\(\therefore \) The gradient is 2
Example 3
Solution
The line touching this curve is the tangent.
The gradient of the tangent can be found by finding the first derivative of the equation of the curve.
The above derivative is the slope of the tangent of the curve at the referred point.
The slope of the tangent at the point (1,6) is as follows:
\[ \begin m &= 3*(1)^2 + 4(1) – 5 \\ m &= 3 + 4 – 5 \\ m &= 7 – 5 \\ m &= 2 \end \]
\(\therefore \) The gradient of the tangent is 2
Example 4
Cheryl draws two parallel lines and the equation of one line is 2x – y + 5 = 0. Find the gradient of the other line.
Solution
The given equation of the line is 2x – y = 5
Further, the gradient of the two parallel lines is equal.
Let us find the gradient of this line.
Comparing this with the slope-intercept form of the equation y = mx + c we have m = 2
The gradient of this line is 2
Hence the required gradient of the parallel line is m = 2
\(\therefore \) The gradient of the parallel line is 2
Example 5
The teacher asks Sam to draw a set of perpendicular lines and to write the slope of one line as 2. Help Sam to find the slope of the other line.
Solution
The slope of the given line is \(m_1 \) = 2
\(\therefore \) The slope of the line is \(\frac<-1><2>\)
Interactive Questions on Gradient
Here are a few activities for you to practice. Select/Type your answer and click the «Check Answer» button to see the result.
Gradient Of A Line
Here we will learn about the gradient of a line, including how to find the gradient of a line from a graph, and from two coordinates, and state the equations of horizontal and vertical lines.
There are also worksheets based on Edexcel, AQA and OCR exam questions, along with further guidance on where to go next if you’re still stuck.
What is the gradient of a line?
The gradient of a line is the measure of the steepness of a straight line.
The gradient of a line can be either positive or negative and does not need to be a whole number.
The gradient of a line can either be in an uphill (positive value) or downhill direction (negative value)
What is the gradient of a line?
What is the gradient of a straight line?
How to understand the gradient of a line
Imagine walking up a set of stairs. Each step has the same height and you can only take one step forward each time you move. If the steps are taller, you will reach the top of the stairs quicker, if each step is shorter, you will reach the top of the stairs more slowly.
Let’s look at sets of stairs,
The blue steps are taller than the red steps and so the gradient is steeper (notice the blue arrow is steeper than the red arrow).
The green steps are not as tall as the red steps so the gradient is shallower (the green arrow is shallower than the red arrow).
Gradients can be positive or negative but are always observed from left to right.
The linear relationship between two variables can be drawn as a straight line graph and the gradient of the line calculates the rate of change between the two variables.
Currency gradient example When calculating the exchange rate of two currencies, we can calculate the gradient of the line to find the rate of change between them.
Here, the exchange rate between pounds ( £ ) and dollars ($) is equal to \frac <3>
Gradient formula
The gradient formula is a way of expressing the change in height using the y coordinates divided by the change in width using the x coordinates.
So using the gradient formula to find the gradient of a straight line given the two coordinates ( x ₁, y ₁) and ( x₂, y₂ ), we need to work out:
This gives us a gradient formula of:
It can be helpful to think about this formula as: ‘Change in y divided by change in x ’ Or ‘Rise over run’
Gradient equation
The gradient equation is another way we refer to the gradient of a straight line using x and y coordinates. So again the gradient equation is seen as m = rise / run where m is the gradient or slope.
How to find the gradient of a line
To find the gradient of line you divide the change in height ( y₂ − y₁ ) by the change in horizontal distance ( x₂ − x₁ )
Gradient of diagonal lines
Let’s have a closer look at the gradient of 4 lines
How far apart do the coordinates we choose need to be?
Top tip: Use two coordinates that cross the corner of two grid squares so that you can accurately measure the horizontal and vertical distance between them. Use integers as much as possible!
Remember: the change in x is horizontal, the change in y is vertical.
Gradient of horizontal and vertical lines
There is no relationship between x and y on horizontal or vertical lines and so they cannot be written in the form y = mx + c as the gradient cannot be measured.
Let us look at a couple of examples to further understand the equations of horizontal and vertical lines.
Example 1
Example 2
How to calculate the gradient of a line
In order to calculate the gradient of a line:
Explain how to calculate the gradient of a line
Gradient of a line worksheet
Get your free gradient of a line worksheet of 20+ questions and answers. Includes reasoning and applied questions.
Gradient of a line worksheet
Get your free gradient of a line worksheet of 20+ questions and answers. Includes reasoning and applied questions.
Gradient of a curve
We can estimate the gradient of a curve at a given point by drawing a tangent line at that point and calculating its gradient.
A tangent line touches the curve at one point only. For the purposes of GCSE Maths, the tangent line is an estimate drawn by eye, but you should try to be as accurate as possible.
Once the tangent line has been drawn in, use the method described above to calculate the gradient of the tangent line. This gives an approximation or estimate for the gradient of the curve at that point.
Related straight line graphs lessons
This lesson is part of the bigger topic, straight line graphs. It may be helpful to take a look at the topic page, straight line graphs before moving on to the more detailed individual related lessons below:
Gradient of a line examples
Example 1: using a straight line graph (positive gradient)
Calculate the gradient of the line:
Example 2: using a straight line graph (negative gradient)
Calculate the gradient of the line:
Select two points on the line that occur on the corners of two grid squares.
Example 3: Using a straight line graph with two coordinates (positive gradient)
Calculate the gradient of the line:
Select two points on the line that occur on the corners of two grid squares.
Sketch a right angle triangle and calculate the change in y and the change in x.
Example 4: using a straight line graph with two coordinates (negative gradient)
Calculate the gradient of the line:
Select two points on the line that occur on the corners of two grid squares.
Sketch a right angle triangle and calculate the change in y and the change in x.
Example 5: given two coordinates (positive gradient)
Select two points on the line that occur on the corners of two grid squares.
Sketch a right angle triangle and calculate the change in y and the change in x.
Example 6: given two coordinates (negative gradient)
Select two points on the line that occur on the corners of two grid squares.
Sketch a right angle triangle and calculate the change in y and the change in x.
Common misconceptions
The change in x is x₂ − x₁
The change in y is y₂ − y₁
When you subtract one coordinate from another, one or both of the numerator and the denominator can be negative. If one is negative, the gradient is negative.
If both are negative, remember a negative number divided by another negative number is a positive number, so the gradient is positive.
