okay guys let's take a look at lesson three vertex form with parameters p and q i don't mean to jump around but we're going to start on the page eight where it says y equals x squared we want to investigate that it says for any number x it's square y is a function of x this is a quadratic function with equation y equals x squared so what i want to do is i want to just create a simple little table of values where i'm going to put numbers in around the origin so i'm going to put in values of negative 3 negative 2 negative 1 0 1 2 and 3. and if i do that i take those values and i square them negative 3 squared is of course nine negative two squared is four negative one squared is zero is one zero squared is zero and then we get one four and nine so when we plot those points negative three nine negative two four negative 1 1 etc what we get is we get not a linear function right because in grade 10 we were to study quite extensively y equals mx plus b which is a function with a degree of one okay the degree here being what little one right there and that's a linear function okay so now in math 20 we're really looking at a degree of 2 okay a function with a degree of 2 and this is a quadratic function and now this is our basic quadratic function of y equals x squared sometimes i've heard it called like an identity function um so that's what another thing we could call it just the basic most simplest raw quadratic function y equals x squared okay now since the graph of every quadratic function is called a parabola the curve is called the parabola the characteristics of the parabola are listed below label them on the graph so the vertex we can see the vertex the vertex is like the summit either a lower or an upper summit so our vertex here is zero zero okay there's our vertex of zero and zero our maximum value well there is none because the graph goes up to infinity and beyond okay our minimum value when i talked about the minimum value that's like the minimum if i said what's the minimum of your value the minimum height if i say what's the minimum you're going to grow the minimum value is like the height so we're looking at the y and the minimum y is a zero okay so my minimum is zero when x equals zero okay and here's our minimum we can label that as well minimum okay the axis of symmetry and actually i'm going to actually change this this should just say equation of axis of symmetry okay and the equation of the axis of symmetry is where would we break this graph in half and fold it in half we would fold it in half right down the middle through the vertex the equation of that that line that horizontal sorry vertical line the equation of that vertical line is x is equal to zero and that's the equation of the y-axis okay the equation of the x-axis incidentally is y is equal to zero but we're looking at this vertical line here the equation there is x equals zero okay my x-intercepts well there's just one right here on the x-axis so there's the x and the y intercept okay which are 0 0 and 0 0 the domain if we look at the domain the smallest x value if we think about this graph the left and rightness is it's going to go forever to the left and forever to the right so our domain is the set of all x's such that x is all real numbers and our range is the set of all y's such that y is greater than or equal to zero right our smallest y value here is zero and then we go up to infinity so our x we go to negative infinity and positive infinity and our range we start at zero and we say y is greater than or equal to zero okay now we'll look a little bit more about with this after we haven't had to look at this these are just some of the things that i'm talking about here the graph of a quadratic function is a parabola here's some examples of what they look like etc etc we'll come back to that in a bit so let's just explore it says investigate for the following equations complete the table of values plot the graphs points using a different color and for each equation complete the chart below so we've already done this with y equals x squared this is 4 1 0 1 and 4. so i know a trick for how we can graph this i always want at least 5 points to graph my parabola so once i know where my vertex is which is at zero zero for this particular graph i know that to find my other points i can move left one right one and right one up one left one up one right one up one i can move left two up four and right two up four and the reason i do it like that is because there's a pattern and i know that if i'm moving right one one squared is one if i'm moving right 2 2 squared is 4. if i want to find the other points if i move right 3 3 squared is 9. and the reason i want to introduce you to the pattern is because we're going to be moving graphs around moving our vertex around but once we move it we don't have to look at the individual points because we understand what the graph kind of looks like like so there's our y equals x squared let's look at the next uh equation we've got y equals x minus 2 all squared so in this case here for this y equals x minus 2 all squared x has been replaced with x minus 2. so what do we think might happen to the graph well i've had students say it looks like the graph might go left too because it's x minus two it looks like the we might move up two or down two so there's all kinds of guesses let's figure out what happens so because x has changed i'm replacing x with x minus 2 what i want to do is i want to look at another point on my graph so i'm going to take like i can take any point i'm going to take 2 4 and i want to leave the y value the same and i'm going to sub that in when y is equal to 4 on my original graph what happens to x okay up here when i y is equal to 4 x is equal to plus or minus 2. so what about down here if i take x to replace x with x minus 2 what happens to that what happens to my function so let's do that now if y is equal to 4 i'm going to sub that in into my y position and i get x minus 2 all squared what i want to do is i want to solve for x to see what happens to my individual points so when i take the square root of both sides right of course that's how i i want to isolate x so i take the square root when i take the square root of a number though i know there's two answers so i have to do a plus or minus the square root of 4. just as a reminder what i mean by that is if i have something like x squared is equal to 16 for example well there's not one value there's two values we can put in for x that will give us 16. we can put a positive 4 squared is equal to 16 or a negative 4 squared is equal to 16. so to ensure that i don't forget one of those values when we take the square root of both sides what we do is we have to take the plus or minus the square root of 16 so x is plus or minus the 4. so the same thing down here we're taking the positive and negative square root of 4 which is plus or minus 2 and that gives me x minus 2 equals plus or minus 2. i want to solve for x i'm going to isolate the x minus 2 isolate the x by moving over the minus 2 and that becomes a positive 2. so we end up here with 2 plus or minus 2 okay let's move the 2 here 2 plus or minus 2 equals x so x is equal to 2 plus 2 and x is equal to 2 minus 2 whoops which is zero which i'm thinking that faster than my hand can write and two minus two is zero so now instead of our points being negative two four and two four now our two coordinates are zero four and 4 4. so where is that 0 4 is right here and 4 4 is right here what has happened to our graph our graph if we were to plug in even if we plugged in the 0 0 if y is 0 and we want to solve for x square root of zero is just zero so x would be equal to two so what's happened to the vertex the vertex is now two zero look at that information here's my new vertex it's moved over two to the right okay so we've moved right to how can that be that's how it can be now if i know my vertex is here at two comma zero then i know i want to go over 1 up 1 over 1 up 1 over 2 up 4 over 2 up 4 over 3 up 9 and left 3 up 9. okay and there's i've actually got my seven points right we kind of like at least five but there's our seven points for y equals x minus two all squared what about x plus one squared if x is replaced with x plus one if we were to put that in our calculator or solve we would see that we end up going left one okay so let's do that in maybe a red color okay our graph is going to move left one and again over one up one over one up one over two up four left two up four right three up nine and left three up nine and there is my new graph now note all of these are the same shape we have just shifted the graph left or right to or right to or left one okay if we were to look at the coordinates here for the blue graph that we moved to the right two this becomes four this is one zero one and four so all of our y coordinates have stayed the same it's the x coordinates uh have changed and same here this one here uh this is when x is here regularly when x is negative two y is four here when x is negative three y is four so our graph has moved left one okay and we don't have to necessarily ice investigate each individual point but that's the idea now uh we've already filled in the first graph i'll fill in the second one and then i'm going to give you guys a second to i'm going to pause the video and let you fill in the next one so our vertex here our vertex here was two comma zero let's do our blue graph first two comma zero our maximum there's no max but our minimum was is zero so our minimum comes from where does our minimum come from our minimum comes from the y value of our vertex right we can see it right here on our graph our minimum is zero when x equals two okay the y intercept for this blue graph was zero comma four the x intercept was two comma zero the equation of the axis of symmetry so let's just do that maybe in a light blue color and that goes right down here right right down through where x is equal to two and again um where's this information coming from or as a hint here here the equation of my axis of symmetry oops i meant to do that in highlighter comes from the x coordinate here of my vertex okay that's where the equation of the axis of symmetry comes from so that one there is x is equal to 2. and then our domain for all of these all of our parabolas is going to be the set of all x's such that x is all real numbers and then our range is the set of all y's such that y is greater than or equal to zero for this particular graph okay so i'm going to pause the video and let you complete the first one here and maybe the third yeah we'll just pause that for a second give those a try if you can okay okay so hopefully you had a second or a minute to try that um and we're able to complete those values there in general it says the graph of y equals x minus p l squared is the graph of y equals x squared moved along the x axis okay moved along the x-axis it was left p units if p is less than zero so if p is negative we go left so up here this is a p is negative p is negative 1 here because this is y equals x minus negative 1 squared right when we make this into a form with a negative the p is the opposite of what we saw here so example this is a positive one there plus one we the p value is actually a negative and that comes from x minus negative one okay if we look at uh x minus two well that's the same as x minus a positive two right if i say i've got 7 minus 3 that's the same as 7 plus negative 3. so this here we're moving right when p is positive p is greater than 0. okay here my p value p is positive two okay because the minus comes from that equation okay so hopefully that makes sense it gets a little confusing because we're going back and forth there the next one says determine the vertex and the p-value so here our vertex is four comma zero and our p-value is four okay because this is x minus positive four squared that's another way of thinking of that okay this one we can think of this is x minus negative six all squared our vertex is of course negative six and zero and our p value is negative six okay our p value is the first part of our vertex the x coordinate of our vertex okay y equals x minus negative 10 all squared our vertex is negative 10 0 and our p value is negative 10. okay and then this last one of course our p value right we've subtracted nothing this is the same as y equals x minus zero squared so our vertex here is zero zero and our p value is zero okay note the p-value comes uh before this oh after after the subtraction sign inside the brackets the p-value is always the opposite sign than the number in the brackets the p-value move the moves the parabola along the x-axis or the horizontal axis okay this is the next investigation for the following equations y equals x squared y equals x squared plus 3. again we know this one this is 4 1 0 1 and 4. if i take those values now my y equals x squared values which i have over here and i add 3 to all of them we get 7. uh 4 3 4 and 7. so what's going to happen to our graph we know our original function let's just pop it out here in green over 1 f up 1 over 2 up 4 left 2 up 4 up 9 up 9 okay over 3 so there's our original function y equals x squared again and then what's going to happen here now instead of negative 2 forward negative 2 7 so let's see negative 2 7 whoops negative 2 7 is right about there okay instead of negative 1 1 or negative 1 4 negative 1 4 and 0 3 and 1 4 and 2 7. so what's happened to our graph the whole function has moved up three okay we can see that on our graph okay and the whole thing is the exact same shape it might look a little skinnier just because we have a smaller portion of it okay but it's the same shape this is our y equals x squared plus three okay and now let's look at the next one let's do a new color here we haven't done any pink y equals x squared minus two so we're going to take all of these x squared values four one zero one and four and we're gonna subtract two so four minus two is two one minus two is negative one zero minus two negative two and so on okay 4 minus 2 is 2. so we're just subtracting 2 from all of our y values and of course our whole graph is going to move down 2. now if i recognize that 0 negative 2 is my vertex which it is and once there i could either move all of these individual points down to okay each individual point or i can say well i know my vertex is here at zero negative two and i know i'm going to go right one up one right two up four if i wanted i could go right three up nine which would take me up to the coordinate of seven right here okay that's the first half of my parabola and the same thing here left one up one left two up four from my vertex or left three up nine which takes me to negative three seven okay so there's y equals x squared minus 2. and of course they want us to fill all this in so i'm going to pause the video and let you complete we've got this one this is the same um let's do that in green this is the same as uh we've done a couple of times so you can copy that over but let's look at the next two i'm going to let you give you five seconds here so you can pause the video and try the next two okay so hopefully you had a minute to fill in these values you may have gotten a little bit stuck on the x-intercepts here for this graph so what we did what i did is i just solved okay these values here i said well when y is 0 what are these values and that there is the square root of 2 and 0 that's the point of the x intercept and this other one is negative root two zero leaving those as an absolute value you could put those in your calculator as well but there we go um y equals x squared minus two i let y equal to zero i isolated the x squared by moving over the negative two which becomes a plus two and then i had two is equal to x squared and i took the square root and got a plus or minus square root of two so those are our x-intercepts the rest i think is probably fairly straight straightforward as long as you're following along and understand okay in general the graph of y equals x squared plus q is the graph of y equals x squared moved along the y axis up q units if q is positive or greater than zero down q units is if q is negative or less than zero okay let's look at the next page um this says investigate y equals p squared plus q so now we're combining the two okay i'm not going to draw y because x squared here quite yet but i know because i know we've got that we've done that many times already three times here we i know i'm going right to the opposite of what the sign looks like and here we're going to go up 3 okay so if i'm looking at this first graph my new vertex when my vertex is normally zero my new vertex is going to be two three okay my new vertex is going to be two comma three moving from zero zero to two three so where's that new vertex two three is right here okay and now i know how to draw this thing i don't even need to look at the other points i'm going to move right one up one left one up one from the vertex right two up four i'm literally just counting one two one two three four right three one two three up nine one two three four five six seven eight nine right we'd be off the grid there okay left one up one left two up one two three four okay so because i held that down i accidentally uh made that so i'm gonna get rid of these little squiggles but you get the idea that's literally what we're doing we're just counting okay we can all count spaces okay and left three end up nine so our graph would look something like this right it's moved and of course we could take the individual points but we don't really need to okay here when x is zero y is seven here when x is one y is uh four okay here when x is two y was three there's our vertex right when x is 3 y is 4 and when x is 4 here y is 7. okay so there's our points i just looked at them off the graph okay but much easier once i know that shape to just move along from there let's try the next one here uh why don't you guys try the next one okay why don't we fill out this down here i know the vertex comes from here my vertex let's fill out this first table together and then the first row here this is the same again the y equals x squared okay but this my vertex is two comma three which we can see there the max where's the maximum come from the maximum comes from the y value of the vertex right how tall are you going to get or the minimum right in this case there is no max okay but there's a minimum so our minimum is three when x is two okay is there a y intercept our y intercept is right there at zero comma seven so we're just looking at the graph the equation of the axis of symmetry well that of course would be right down the middle where x is equal to 2. there's our x equals 2. and remember the equation has an equal sign you don't want to just say 2. okay another way to write this by the way you would say x equals 2 or x minus 2 equals 0. sometimes you see that domain instead of all x x is all reals range y is greater than or equal to three okay i'm going to pause the video and let you guys complete this one and the table below okay okay then so in order for me to fill in this table i started with the vertex i knew i was going left one and down two so i knew my vertex was negative one negative two okay so i started there right on my graph right there vertex negative 1 negative 2 and then i drew the graph from there and then i got the coordinates of the points okay there's a couple of ways you could have done that but that's probably one of the easiest or the quickest and then i filled in the table down here my vertex negative 1 negative 2 my min negative 2 when x equals negative 1 etc so you can check your own answers there if you need to pause the video the last thing is we've kind of got this general formula so we know that uh for this vertex our vertex is always p comma q okay whatever our p value is and our q value our maximum or our minimum where does that come from well our maximum comes from the q value okay so our max is q or we could say max is equal to q when x equals p and then our y-intercept depends okay it depends on the graph okay we can't figure that out so much in this case okay um the equation of the axis of symmetry well that comes from where our uh line of our line of the axis of symmetry would intersect the x-axis okay so here the equation of the axis of symmetry was x equals negative one and here the equation of the x axis of symmetry was x equals two so what's the p-value of our okay it's the p-value of our vertex so we would say here x is equal to p or x minus p equals zero okay the domain is always no matter what all real numbers and the range is going to be y is greater than or equal to q okay so there we are in a nutshell vertex form summary so describe how the graphs of the functions in each pair are related then determine the vertex the equation of the axis of symmetry the domain and range any intercepts and sketch the graph of the second function in each pair so we're going to do a basic we may not cover all of that right now but we're going to do a basic um now the first thing is the description what has happened here the description x has been replaced with x plus two therefore we've got a horizontal shift and call it a horizontal shift two units left okay and then this q value q is equal to negative four therefore we are going vertical shift four units down okay that's our description then determine the vertex the equation of the axis of symmetry etc so our vertex here right i know that if i think of this as y equals x minus negative 2 squared minus 4 my p value is negative 2 and q is negative 4. so our vertex is negative 2 negative 4. okay the equation of the axis of symmetry will come from here x is equal to negative 2 or x plus 2 equals 0. there's the equation of axis of symmetry okay the domain and range the domain is all real numbers and the range y is greater than or equal to negative 4. okay i'm not going to worry about the intercepts right now but if i were to sketch the graph of this function you know let's just do a mini sketch nothing fancy uh where are we negative 2 negative 4 so i'm going to go like this okay negative 2 negative 4 i know that my vertex is right here if i had a nice grid whoops i didn't mean to bump that i would move over one up one over two up four left one up one left two up four and there would be a basic sketch okay and actually i think we can find the x-intercept the x-intercepts are from negative two we move left two and up four so that is the point negative four zero and this is the point zero zero so of course our x-intercepts is negative four and zero okay negative four and zero and our y-intercept is equal to zero right here okay we'll omit this one we've done enough for today so that's great i would suggest you do all of the questions on this uh in this assignment that would be awesome thank you