hi welcome to this cobra's video on the transformations of graphs in this video we're going to look at the transformations of the trig graphs or the trigonometric graphs now before you watch this video it can be useful to watch four of the videos in corporate maps they are the transformations of graphs video which go through the four transformations needed at the gcse level also watch the videos on the trig graph so the y equals sine x graph the y equals cos x graph and the y equals tan x graph and those videos give you an idea as to what those graphs look like okay so first of all just let's just recap what our transformations are at gcse level that we need to know we've got in blue our reflections and we've got in red our translations so our reflections we've got y equals minus f of x that transformation will reflect the graph in the x-axis so the points above the x-axis will go below the x-axis and the points below the x-axis will go above the x-axis and the points on the x-axis will stay where they are they'll be invariant then we've got our y equals f of minus x that's a reflection in the y-axis so the points on the right-hand side of the y-axis will reflect to the left-hand side and vice versa and the points on the y-axis will be invariant they'll stay where they are then we've got our translations so we've got y equals f of x plus a where the plus a is outside of the brackets that'll move the graph a units upwards so if it was plus one the graph would move one upwards if it was minus three it would move three downwards and so on and then finally we've got y equals f of x plus a and that will translate the graph a squares to the left so all the points on the graph will move a to the left okay so they're the transformations we need to know gcse level let's just have a quick recap of the trig graphs so we've got y equals sine x y equals cos x and y equals tan x and these are all in degrees at gcse level they'll all be in degrees so we've got our y equals sine x and i've sketched it between 0 and 360 degrees so as you can see it starts to the origin goes up to 90 degrees and one goes down to 180 degrees and zero 270 degrees or minus one up to 360 degrees and zero and i'll just carry on and we could also draw for the negative values of x also and that they would just carry on going down and up and so on a y equals cos x graph well it starts at 0 1 then reaches 90 degrees and 0 180 degrees and minus 1 and then it starts coming up again and it just carries on and then we've got a y equals tan x graph and that's the graph of the asymptotes those lines where the graph never reach so it starts off with the origin curves upwards we've got our asymptote then after 90 degrees it comes upwards reaches 180 degrees and zero and that curves upwards and just carries on like that okay so let's have a look at our first question so first question says here's the graph of y equals sine x for values of x between minus 180 degrees and 180 degrees so here's the san x graph and we've been asked to sketch y equals minus sine x so as you can see that is our first transformation here our y equals minus f of x so that is a reflection in the x-axis so it flips the graph over vertically now whenever i'm doing a question like this with trig graphs what i tend to do is focus on the key points so points such as the minus 180 degrees and 0 minus 90 degrees and 1 this point are 0 0 and so on and i consider where they go after we apply the transformation so we're reflecting this graph in the x-axis so the points on the x-axis will be invariant they will stay where they are 180 degrees in zero zero zero minus 180 degrees and zero the point is above the x-axis this 90 and one will reflect below so become 90 and minus one and the point below the x-axis our minus 90 degrees minus one will reflect above the x-axis to reflect up to there then all we need to do is draw a smooth curve through those points and that will be the graph y equals minus sine x and i'm going to cheat there we go so the graph would look something like that and that is the sketch of our graph y equals minus sine x okay our next question so next question we've been asked to sketch y equals sine of brackets minus x so as you can see the minus sign is inside the brackets so it's going to be this transformation here our reflection in the y-axis so the points on the left-hand side of the y-axis will go to the right and the points on the right will go to the left and the points on the y-axis will stay where they are okay so let's have a look at our graph so we've got our sine x graph and we know the points on the y-axis would stay where they are so that the point at 0 0 will remain where it is the point at 90 degrees and 1 that'll be reflected to minus 90 degrees and 1 so that will go there the point at 180 degrees in zero well that'll be reflected to minus 180 degrees and zero the point that's at minus 90 minus 1 will be reflected to be 90 degrees and minus 1 there finally the point that was at minus 180 degrees 0 will be reflected to 180 degrees 0. so the key points would look something like that and then all we need to do is draw a nice smooth curve through it and it would look something like that so that is the graph of y equals sine of minus x okay so our next question so in this question we've been given the graph of y equals cos x for values of x between 0 and 360. and we've been asked to sketch y equals cos x plus one so as you can see we've got our plus one and it's not in the brackets so it's just adding one to the whole answer of cos x so that will move the graph one up vertically so it's going to translate the graph one upwards so let's have a look at our key points so the key points in this graph are zero one well that's going to move up once it's going to become zero two the point ninety zero will move up to ninety one the point that was at 180 degrees minus one will move up to 180 degrees zero the point that was at 270 degrees zero will move up to 271 and finally the point though is at 360 degrees one will move up to 360 degrees and two okay and then if we just draw a nice smooth curve through them our graph would look something like that so that would be the graph of y equals cos x degrees plus one so next question says here's the graph of y equals cos x so we've got the cos x graph again and this time we've been asked to sketch the graph y equals cos x minus three and our minus three is outside of the brackets so again it's going to be a translation so it's going to be this transformation here and instead of moving three squares upwards because it's minus three it's going to move three squares downwards so our graph will move three down so the point at zero one would move down if we take three away from one we get minus two the point at ninety zero would move to ninety minus three the point at one hundred and eighty degrees minus one we've moved to 180 degrees minus four the point at 270 degrees zero will move down to 270 degrees minus three and finally the point that was at 360 degrees one we moved down to 360 degrees minus two so the key points would be in those positions so all we need to do now is draw a nice smooth curve through them and it would look something like that okay next question we've been given y equals cos x again now this time we've been asked to sketch y equals cos x plus 180 degrees so we've got our plus 180 inside the brackets so it is our translation horizontally so we're translating the graph if it's plus a in the brackets it moves it a squares to the left so we're going to be moving this graph 180 degrees to the left so if we have a look at our key points well our point is zero one well that's going to move 180 degrees to the left so that's not going to be on the axis so i'm actually going to start at this point here are 180 degrees minus one now with this translation we're moving at 180 degrees to the left so it's gonna move it from 180 degrees to zero so it's going to move to here okay our next point that was at 270 degrees zero we're moving at 180 to the left so whenever we take 180 away from 270 it brings us to 90. it's going to be 90 0. the point that was at 360. well if we move 180 degrees to the left it will move to here so as you can see the shape of our graph is starting to form here so if we carried on our normal cos x graph we would have 450 and zero well if we move that 180 degrees to the left it moves to 270 degrees in zero and our last point will be here and as you can see if we draw a nice smooth curve through them it would look something like that okay our next question now our next question we've been given the graph of y equals cos x so we've got the cos x graph again and we've been asked to sketch y equals cos of x minus 90 degrees so this time we've got minus 90 inside of our brackets so if we have a look it's going to be a translation and so normally if it's plus a we move a square to the left because it's a minus we're going to be moving it to the right so if we have a look at our graph so if you have a look it's minus 90 so that means we're going to be moving all the points on our graph 90 to the right so the point that it was at 0 1 we moved to 91 the point that was at 90 00 would move to 180 degrees zero the point that was at 180 degrees minus one would move to 270 degrees minus one the point there was our 270 degrees zero would move to 360 degrees and zero and so on as you can see we haven't got a point here for zero but as you can tell from the shape of the graph you can figure out it's going to be at the origin but if we just went back to 90 degrees it cos of -90 is zero and if we move it 90 to the right it would bring you there and if we draw a nice smooth curve for it i'd like to example because we haven't looked at tan and i don't want to leave tan out here's a graph of y equals tan x and we've got the tan x graph between 0 and 360 degrees so you can see we've got asymptotes at 90 degrees and 270 degrees so we've been asked to sketch the graph of y equals minus tan of x so that's going to be a reflection because we've got our minus sign and it's in front of the function it's in front of the tan so that's mean it's going to be a vertical reflection it's going to be a reflection in the x-axis so all the points above the x-axis will go below the x-axis and the points below the x-axis will go above the x-axis and the points that were on the x-axis will remain invariant so our zero zero or 180 degrees and zero and our 360 degrees of zero will remain where they are and the graph will come down and it would look something like this whenever we sketched it so it'll go below there'll be a mirror image above there and below a reflect like that and that's it so with the transformations of trigraphs very important you know the transformations so watch that video on corporate maps if you need a recap so make sure you know the four transformations at gcse level our reflection vertically our reflection horizontally our translation vertically up and down and our translation horizontally okay so know those four transformations and it's also useful to know your trig graphs to know what the 10x graph looks like the cos x graph and the tan x graph and then it's just a matter of applying those transformations and whenever you're applying the transformations it can be really useful to consider the key points and consider where they move whenever you apply the transformation and that's it