we will start this video by looking at the grade 11 parabola so let's start off how do we know if a equation that has been given to us is a parabola we should recognize the following y equals to x squared the important part that makes something a parabola is that part over there please note that that is very similar to a exponential graph which looks as follows and here on the right we have an exponential graph the main difference is that an exponential graph has the x in the exponent whereas the x is the base for a parabola so please make sure that you do not confuse those two the equation that you can see would have a graph that looks as follows that is a standard parabola where the turning point is at 0 0 we can now modify this parabola to make it move upwards and downwards as we saw in grade 10 but now in grade 11 we can also make it move left and right let's see how we would have to modify the formula to do that having a plus one written in the formula causes the graph to be shifted one unit upwards and so the new graph would look as follows and so now we can see that the gra the original parabola has been shifted one unit upwards but all the rest the the shape of the graph and everything else stays the same so these coordinates were 0 0 these coordinates would be 0 for the x value and one for the y value because it has shifted one unit upwards we are now going to look at the new feature for grade 11 which is being able to translate a graph we are now going to look at the new feature for grade 11 which is the ability to transfer a graph horizontally we will do that as follows and there we can see the new or the new part for grade 11. to show that a graph has been shifted horizontally you go to where the x value was and you modify the x value only so the way you show that is by putting it in brackets okay now one extra thing that's very important with horizontal shifts so shifts to the left and to the right you have to think in opposites so saying -2 does not shift the graph left it actually shifts the graft graph right so the resulting graph from that equation would appear as follows here we can see a rough sketch of what that parabola would look like let's analyze it carefully we know that from this part over here it's a horizontal shift which is the new feature for grade 11. x minus 2 means it's actually going to go two places to the right so there it's gone two places right remember the turning point started over there so it's moved two places to the right and then the plus one moves it upwards and so this is the new turning point okay so that is what that graph will look like over there there is one extra feature in the parabola and i'm going to explain that now so we've looked at what this number does we've looked at what this number does but we haven't really understood this number all that that number does is tell you whether the graph is happy or sad so whether it looks as follows or it could look as follows if a is or if this number in the front is positive then it's a smiling graph if that number is negative then it's an unhappy graph the value itself simply tells you whether the arms are far apart or close together so that will be shown as follows so what i have tried to depict in these two diagrams is that the larger the number in front of the x squared the more narrow the graph is or the more closer the arms are together but that really you guys aren't going to have to worry too much about what that number does i'm just explaining its purpose but they're never going to try to test you on that or anything like that so don't worry too much it's just out of interest i'm now going to explain one more thing about that number that i've just mentioned the important thing about that number is that if it is negative then the graph will look unhappy so negative unhappy if it's positive then it's happy smiling that's a more important part of that number that you do need to know whether it's a two or three that doesn't really matter now that we have a better idea of what each part of the equation does to the graph let's get into a good habit or let's practice how to do a sketch graph of the parabola please note that this example where this activity is not intended to take long it's not you don't have to worry about finding the exact x-intercepts or y-intercept all that we are practicing in this activity is whether you know that the graph has moved up down left or right and whether it is happy or sad in the next part of future videos we're going to look at the specific details of how to draw the graph we'll start with number one so number one is definitely a parabola because it's got x squared the x is not the exponent it's a happy x parabola because the number in the front is not negative and then this minus 4 because it's not in brackets with the x it means it's it's not a horizontal shift it's a vertical shift so it's moved four units down and so that graph would look as follows and so there we can see that graph sketched note that the turning point which originally starts at 0 0 has not moved left or right it has only moved down okay we don't know what the exact x-intercept values are but we are not interested in that in this activity let's start with number two for number two we can see that it's definitely a smiling parabola because the number in the front of the bracket or in front of the x sorry is not negative it has moved one unit to the right remember the horizontal shifts are opposite minus means to the right plus means to the left and then it has also moved four units down and so the new turning point will be located as follows and there we have the parabola because you must remember that the turning point always starts at 0 0 originally it was then moved one place to the right and then four units down and so the new turning point is one and minus four once again we don't know exactly where this value this value or this value is but as i said in the future examples we will practice how to find those points we can now go to the next example for number three let's and we can analyze the various parts of that equation we now see that it's a negative meaning it's going to be a upside down parabola it's been shifted one unit to the right because it says minus one which means to the right and it's been shifted four units upwards so without having to worry about this being plus or minus we can still go and locate the turning point by moving one place to the right and four units up and there we can see that parabola drawn it's a sad parabola shifted one place to the right and four units up with number four the number in the front is positive so it's a happy graph this plus four means that it's actually gone four units to the left and this plus three means three units up so it's going to be four units left and three units up and so that is what that graph would look like there the turning point which always starts at zero zero gets moved four units left three units up and it's a happy graph and now we can start with number five analyzing number five we can see the number in front of the bracket whenever i say that's positive it's because there isn't a number there at the moment and so we know that that's just an imaginary one which is positive the minus six tells us that the graph has shifted six units right not left it must be to the right because it says minus opposite to what you would naturally think and then two units up so the turning point will be located at six and two and then it's a happy graph the last part of this video will will talk about how to correctly draw a parabola so how to do it the proper way and then in the next video we'll do many examples so find drawing a parabola is a three-step process you will need to do or you'll need to have the following three things you'll need the x-intercept the y-intercept and the turning point to do the x-intercept you make y equal to zero to do the y-intercept you make x equal to zero and then there will be three ways to find the turning point which i'll discuss with you in this in this example the equation that we are going to use for our example is y equals x squared minus seven x minus five so let's start with step one the x-intercept to find the x-intercept you make y zero you would then try factorize if not possible you could use the quadratic formula and there you have the quadratic formula so you could plug all the values into that equation and if you do that you're going to get two x-intercept values of and so there we have our two x-intercepts which can then be placed on the diagram we could then find the y-intercept by making x zero we could then find the x-intercepts we could then find the y-intercept by making x zero you could then plug that into your calculator and get an answer of y equals to negative five you can then place that on the diagram and then the last step would be to find the turning point in which case there are three methods but in this example we will only be able to use two and i'll show you the third one afterwards so the first method is to find the middle value between your two x-intercepts between this x-intercept and this x-intercept so to do that you just add them together so you add those two values together the minus 0.65 and the 7.65 and then you divide that by 2. it's like taking the average that's one way to do it the other way is to use a turning point formula now that turning point formula can be located within the quadratic formula the part that is highlighted in red is the formula that can be used to locate the x value of the turning point and so let's give the second method a try so you simply go and type in minus b over 2a on your calculator as you would normally use the full formula now you only use need to use a small part of it and that's going to give you an x value for the turning point as so the x value of the turning point is 3.5 so if you have the x value of the turning point it's 3.5 but we now need to find the y value so many people straight away say make y equal to zero but that's how you find x-intercepts so if we know that the x-value of the turning point is 3.5 then we simply plug 3.5 into the original equation to find the y value of the turning point and so there we have the y value as minus 17.25 so now we can go locate those two coordinates on the diagram and now we are at a point where we can draw the parabola so i didn't get all the the lines perfectly connected to the dots but that's okay i'm sure you guys understand the procedure what i now need to do is just explain the last method of finding the turning point which can only be used under certain circumstances so now we're going to talk about the turning point form so remember in the previous slide it was quite easy to locate the turning point that was because those equations were written as follows so when your equation is given like this instead of written in the normal trinomial way then we can get the turning point straight away we know that this is a parabola that has shifted one unit to the right and four units up and so the turning point which always starts at zero zero would also move one unit right and four units up and so the turning point for that equation or for that parabola would be at one and four so we call this the turning point formula for a parabola or the turning point form sorry so if you are given an equation in that form already then you can get the turning point straight away if it is given to you in a trinomial kind of way then you are going to have to use either that method or that method i hope that makes sense in the next video we'll practice some example