okay exploring quadratic relations graphing let's take a look at the example of the mini golf example in exercise 6.1 on page 322 in the textbook so if you have your textbook here you can turn to page 322 and you'll see that example after the ball is hit it travels up and then back down right cuz what goes up must come down the distance the ball is away from the starting point can be given by the quadratic relation Y is equal to -2.5 x^2 + 6x where y represents the blank in blank so what's the Y value when that ball goes up and back down what does it say anything on the graph relation we're graphing ball is hit and what do we where X is the time and Y so the distance of the ball from its starting point in meters right so that's what we're going to fill in here so y represents the distance in meters and X represents the what's X time in seconds that the ball travels okay so we have time running along the xais and the distance going along the uh line graph the relation in the graphing calculator with an appropriate window sketch the graph below label the access etc etc okay so I'm just going to pause for half okay so turn on the calculator uh let's get rid of that so we're going to go to Y really seriously okay we're going go to y equals and we are going to type in -2.5 so that negative is the negative at the bottom it's this guy here right it's not that's a minus sign we don't want the minus sign we want negative 2.5 x that's up here squared uh plus 6X plus 6 x and let's say we just hit graph so if our calculator is sort of manely cleared or whatever you've got a window which goes from -10 to 10 on the x's and -10 to 10 on the Y's and you can sort of say well wait a sec if I'm hitting a golf ball it doesn't make any sense to have negative time right so we'll start the X at Z zero and then well how far out should we go so this is like 1 2 3 so maybe three right cuz it's going to come down and it's going to you know H the ground or the distance of travel will be back there and how high does it go well let's say we go to five so we're going to set our window not on the board apparently we're going to set our window so it doesn't make any sense to have a negative x value because that's time and we don't need to go to 10 seconds because apparently in three and we don't need to have a distance a negative distance that doesn't make any sense either right so we saw what the graph looked like but in the context of the question we only want a piece of the graph and we only want that piece that's in the first quadrant so we're going to put a zero oops I'm going to put a zero in for y Min hang on zero I could do it on the board wait window if you're following along with the video at home don't do this okay zero and I think five is high enough and graph okay there nice graph right it it fills the window there's there's still a little space around the edges so we can see the whole thing so what have we got here we have our window going from Zer to three by ones and we have our window for y going from 0 to 5 by ones now this is how we describe the X window and the Y window right and the way it works is when we say x this is the X Min this is X Max and this is X scale okay and then the same thing here y y Min y Max y scale okay so if we sketch that it looks like this let's uh put these there's three two one okay and that's the X this is the Y and what we got one two three so looks like maybe 3 point something three okay that up a bit okay and if we want we you know one two 3 one two 3 four five so what do we call the shape of a quadratic relation that shape Parabola right yeah Parabola so the shape of a quadratic relation is called a parabol parabola all qu and we're doing these for the first couple of units right so we want to make sure we get down the lingo all quadratic relations have a basic standard equation in the form y = ax2 so a some number * x^2 plus some number which may be the same it could be different so b x + C where the parameters a B and C are real numbers okay so they don't be integers right we don't even have we got -2.5 right so it's real number okay we all good with that good changing the parameters and showing the effects on the graph let's look at a demonstration to see how changing the values of the parameter affects the graph of a quadratic relation answer the following questions at the appropriate time in the demonstration okay so hang on let's just pause for a sec okay so just set the scene so we're on decimal.com um which you can't see uh if you're not here so it's decimos D I can't even write on it okay go back to here www do www.de mod DM o.com okay so I'm going to go here I'm going to type in y = a x^2 plus b x whoops plus BX + C and notice here here it says add slider right and I going say add a slider for all a b and c uh so what that does allows me to change the values of a b and c just by grabbing it and sliding it along on your graphing calculator you would have to regra each time right like you'd have to go in and write y = ax so here's what we're going do let's I well I say l let's we're going to set B to zero and C to zero so we're starting with the graph y = ax2 and a is 1 we're just started with y = right the basic Parabola the basic quadratic relation okay that's the basic quadratic relation um you know reasonably centered on the screen it's got uh this point here 0 0 and what I'm going to do is let's play with the a value and see what happens right so I'm going to take the a value so I can just take this slide at home you would just click on this and then just drag it along right so I'm going to click on this and drag it along Al and what's happening to the graph Getting Thinner Getting Thinner yeah narrower thinner steeper if you like okay and I take a back down I'm not going to take a into the negatives and we're going to say so what happens to the graph when a is what happened to the graph when a is negative yeah it opens down instead of opens up right and if I make it larger negative what's happening to the gra Getting Thinner right or narrower so apparently the larger the a value whether it's positive or negative we get this thinner graph does does the basic shape change no it's still the same shape right still so it's still a parabola right all these quadratic equations or all these what are we saying quadratic so the common characteristics all parab whoops okay so let's set that back to one so now we know what a does right a affects the shape of the graph and a affects the direction of opening right does it open up a is positive it opens up a is negative it opens down okay B come on okay so what's happening with B it's kind of move right it's moving right it's moving it around is the shape changing no still a paraba okay so let's set B to zero and pull it back here and C what's it doing here how's it moving which direction is it moving up up and now it's moving down down shape changing at all no no okay so what have we got so all parabas have that shape right sort of the the curvy shape symmetry so symmetry is it's symmetrical about a line running through its center right this point here so they all have what's called a Vertex this point is called the vertex so it's the lowest point if it opens up and it's the highest point if it opens down okay so that's the vertex let me just shift this up so there's the vertex there's a line that runs through the middle called the axis of symmetry if you want to think about that if I put a mirror on here it would reflect and if I looked in the mirror I'd see the problem right because one side just is a reflection of the other in this line which runs through the middle and has is called the axis of symmetry just got to close that uh so symmetrical about a line about a well it's a it's an up down it's a vertical line right so symmetrical about a vertical line passing through the vertex what the common character they have the same shape same basic shape okay and let's just sort of sketch one out they have a point here that's called the vertex so it's where they turn around right I mean it's going one way it's going up and then it turns and it's heading back down or it's coming down and then it turns it comes back up were all of them functions all the on I played with the ABC did it ever not become a function so they were all functions right for every x value there was only one y value all the quadratic yes what is the degree of the equations what degree were they so we did this right the other we talked about degree we talk about degree yeah we talk so what degree are all quadratic relations two degree two yeah exactly so they are all degree two uh what degree equation of all two that's characteristic every second degree equation every equation of degree 2 will be a parabola will open up or down depending on the value of a will be symmetrical about a line vertical line that runs through its middle and the middle of the parabola it's called the vertex okay this line here is called the axis of symmetry so the vertical line it's called the axis of symmetry right move on if if only the parameter a is changed what effect does it have on the graph so we took a and it made it narrower so makes it narrower as a increases in value what else does a do FPS it right so reflect it so the a if a is positive it opens up if a is negative it opens down so if a is greater than zero it opens up if a is less than zero it opens down so a controls kind of the shape and the the shape is in how it's always the same shape but how how you know steep it is or how wide or narrow it is and also the direction of open if we only change the parameter B what effect did it have when we change B just move the vertex right so moving the vertex and when C was changed what did it do upward down right so if C was greater than zero it moved up and if C was less than zero it moved down right one way to think about that is when you put the x value in and you do the whole ax s and BX and so the a is kind trying to determine which way does it open and uh and how narrow or wide it is and the B sort of moves it a little bit once you've done that if you take that vertex if you add a number on the end so I add 10 then it's going to move up 10 right so that's what C does it moves it up or down so if C is greater than Zero moves up if C is Less Than Zero moves down okay I'm not going into great detail on the B value just because it's really hard to look at that and sort of like I can predict if C is 10 I know it moved up 10 if B is five I can't really say Well it moved that many you know this much that way right it's just got a more profound effect on it so uh if the equation of the parabola is written in the form y ax2 + BX plus C then the parameter blank is the Y intercept of the parabola hang on let's go look here so so here C is 4.1 let's see go away damn it hang on I'll do it on here there C's four uh let me move b a bit what's what's the Y intercept right now where's it intercept four four okay has it changed because I'm changing the B no okay what if I change the a is the Y intercept changing so which parameter is the Y intercept wait no wait which parameter is the what what if the Y intercept just Chang to where the Y intercept just change to yeah so the parameter that's the Y intercept is C okay tell me about the Y intercept if I say Y intercept and you say the Y intercept occurs when when the x value is equal to zero so if we have ax^2 + bx+ C and I make x0 what happens to this term Z what happens to this term so all we're left with is y the Y intercept which is C so the parameter C is the Y intercept right because the Y intercept occurs when X is zero if I make x z the ax s term that's zero the BX term that's zero all I'm left with is y = c see predicting values of parameters for each of the parabolas below predict the values for the parameters B and C if the equation for each Parabola can be written in the form y = x^2 + BX + C write the equation check your answers with the graph and calculator by entering the equation and choosing the windows to match the grid okay so I'm not going to choose the window to match the grid and I'm not going to do it on the graphing calculator right although keep in mind you have to know how to use the graphing calculator because that's the thing you get to bring with you to the tests and that's the thing that you get to use on the final exam and that's the same deal in 30-2 right or 30-1 you know if you go on to that down the road all you get to use on test is the graph and calculator but for exploring this stuff you know we can use this so I'm going to set a to one but actually going up here and typing in a one okay and woo okay I didn't want to do that and let's just here we'll set B to some other value okay so hang on let's go here what's the value of okay so ax2 so how do you know that a is equal to one right so I want you to think about this let's go back I'm going to go back to the basic Parabola so I'll set B and C to zero and we'll just take a look at some of the characteristics here okay so what is it and here I'm going to move this actually I'm going to put this away so we're looking at y = ax2 right 1 x^2 no b b0 c is zero okay I'm going to move this here so we can see a little bit more so this is your basic quadratic function your basic Parabola y = x^2 and so if x x is 0 0 S is 0 right 1 SAR is 1 so you put in an x value of one you get a y value of one 22ar is four 1 2 3 4 3 squar is 6 7 8 9 -3 squar is-3 quantity squared is Right remember it's symmetrical about in in this case it's symmetrical about the y axis right so if I move three units away from the Y AIS here and go up nine then if I move three units this way right to the left of the axis of symmetry how far up do I have to go nine right because it's symmetrical okay so you've got 39 you've got - 39 you've got 4 16 which we can't quite see because it's just a little bit off okay but if I want to get the a value then I know that if I don't mess with a then if I move one unit away I should move one unit up right cuz 1 squ is one so if I look at the vertex and go one unit from the vertex and go up now here let's play with this a little bit so let's say I make a two okay so now I've moved one unit away how far up am I going two two cuz what are we doing right we're saying okay let's remember order of operations right you got to square the x value so let's Square 1 1 squ is 1 then we're going to multiply by a so it's 1 * 2 is 2 so if I move two units away and normally I would go up four but now it's 2x^2 how far up do I go so what do I have to do I took the one and I doubled it made it two I take the four is going to become eight 2 3 four five six seven eight right I think we can actually Trace along here right and we can see oh look there's the0 28 okay if I went three how far up would I go so normally it's how far 3 3^ squ is nine but I've got the two in front so it becomes say it say it loud 18 18 what if I change a to three okay so normally we move one unit from the vertex we go up one unit but how far up are we going to go now three normally I move two units 2^ SAR is four but how far up do I go now 12 right there's the point there 212 okay so that's what a is doing so when I go look at I'm going to set a back to one because we're supposed to just be doing ax2 but what I wanted to show was uh how do I know what a is okay I move one unit I go up one a is equal to 1 how did it get down to -2 which coordinate affected the Y intercept C so what's C -2 what do you think B is for this what did B do it moved it right does that look moved no it only moved down so what's B zero right no movement if there's no side to side right what does this do changes the shape tells me which way it opens right so makes it narrower or wider well on this case how do I know if it's narrow or Y I go over one up one I go over two 1 2 3 4 I go up four right 2^2 is four if it goes up four then the a value is one right if it goes up eight I know the a value is two if it opens down I know the a value is negative if it's moved left or right I know there's a b value and if it's not moved left or right then B is equal to zero okay this one from the vertex I go over one up one I go over two 1 2 3 4 what's a one go over one up one right that's the normal that's the unstretched right a is a stretch factor it stretches it up or down yep I just confus like how we know like what A and C are okay so a from the shape right so a I start at the vertex and I say if I move one unit from the vertex and this thing has not been made narrower right then its basic shape means that I should go up one over one up one a is one if I went over one and up two then I would say Well normally when I go over one I should go up one but I'm going up two that means they is two do that from the vertex