in this video we're going to look at what the term gradient means and see how we can calculate it from a graph using three different methods one where we find out how much the line has risen by for each one that it goes across a second that uses the rise of a run equation and a third that uses the change in y over change in x equation although as we'll see these second two methods are basically the same thing now the gradient is basically just a measure of how steep a particular line is so if we took these four hills this first one in the top left has the highest gradient because it's increasing in height most quickly or in other words it's the steepest meanwhile the second one is less steep and so has a lower gradient the slope in the bottom left though isn't rising at all it's completely flat so this one has a gradient of zero because it's not going up or down and this last one is sloping downwards so we say it has a negative gradient and if it was sloping downwards even more steeply then its gradient would be even more negative if we show these lines properly on graphs instead though then we can actually calculate the gradient of each one but for the sake of space let's move them all over to the side for now and look at them one by one starting with the first as we mentioned at the beginning there are a few different ways that we can find the gradients but the most simple technique is just to figure out how much the line goes up by each time that it goes across by one for example if we pick any point along our line like this one here and we draw little dashed lines going across by one and then up until we meet the line we can see that for every one that it goes across to the right it also goes up by one so the gradient of this line is one and would have found the same gradient no matter where we looked along our line if we look at our next line though and do the same thing this time for every warner that it goes across it only goes up by 0.5 and so the gradient of this line is only 0.5 which means it's less steep than our last line another way to think about the gradient is to use this equation here which says that the gradient is equal to the rise divided by the run with the rise being how much the line has gone up by and the run being how much the line has gone across by you might also have seen it as change in y divided by change in x because the rise is basically how much the y value has changed by and the run is just how much the x value has changed by so these two equations are basically the same thing which means you can use whichever one of them you want so if we use the equation with our example here we just figured out that it went up by 0.5 so our rise or change in y would be 0.5 and it went across by a 1. so that's our run or change in x which means that our gradient would be 0.5 divided by 1 which is just 0.5 just like we got before importantly though we can also use this equation for longer stretches of our graph as well for example if we wanted to find the gradient between these two points which are quite far apart then we need to draw dashed lines between them by going across and then up and then figure out exactly how much we want to cross and up by so if we start with how much we went across we went from x equals negative four all the way to where x equals two so our x value has increased by six then to figure out the rise we went from where y equals negative one up to where y equals two which is an increase of three then we can put these figures into our equation by doing the rise or change in y of 3 over the run or change in x of 6 which gives us 3 divided by 6 so 0.5 again if we switch to our third graph now this one doesn't rise at all so no matter which points you pick along the line the rise will always be zero which means that our gradient will always be zero as well moving on to our last graph one thing to point out is that you always have to think of lines as traveling from left to right so this line is going down and will therefore have a negative gradient to find what that gradient is we can use any of the techniques that we've looked at so far the easiest one for simple graphs like this is to pick any point along the line go across by one and then see how many you have to go up or down by so because we had to go down by two which is a change of negative two we know that the gradient must be negative two for this line to use one of the equations instead we pick any two points along the line and find the rise over run so if we draw dashed lines between these two we can see that it's gone down from three to negative three on the y-axis so a change of minus six and along from negative one to two on the x-axis so a change of three and if we then plug these values into our equation we're going to get negative 6 divided by 3 which gives us a gradient of negative 2. anyway that's everything for this video so hope it all made sense and cheers for watching