Stony Brook MAT 123 Fall 2015
Lecture 12: Lines, circles, and parabolas
October 5, 2015

Start   Okay so the first thing we are going to do is stuff about lines, you guys should be very comfortable with lines, right?
when we graph, we draw a line to based on the relationship we show that relationship with the line because you move a certain amount up and a certain amount across and its always the same amount across or whatever relationship we are looking at okay?
thats what we call slope.
0:31we want to factor a certain line is a line has a consistent slope, every time you go across to x you go up y. okay?
thats why you can use slope on line in other occurs the slope changes, thats what calculus is about, the one slope is changing and we can find the equation of the line if we know two different spots on the line
1:00so im assuming everybody can find the slope of a line with 3 points so lets practice can we use the one from webassign? sure this is something you shouldve done in AMP 103
1:30oh lets see so we have two points, so we know when we represent these, we say 10,3 and 6,11 we say to ourself every time it goes down a certain amount, it goes across a certain amount of visa versa when you get to the slope you can really call it what you want
2:03so when you represent these algebraically, its gonna be y is the x to a single power, 1 and well make an equation for you so whatn do we need? we need the slope slope as you all knw i hope is the change in y over the change in x
2:31or y2 minus y1 over x2 minus x1, it doesnt matter as long as its consistant what this means is for every certain change in y, i have a certain change in x and its always the same on a line as proposed to a parabola circle, fun graph and you can pick what ever points you want to be x1, y1 and x2,y2 so how bout we make these
3:02number 1 these number 2 so this would be 3-11 and 10-6 -8/4 -2 so far thats very straight forward and then we need an equation for a line so many of you love the equation mx+b
3:32but the truth is its better form of that equation that you want to use we potential get from rearranging this this is called the point slope form because it using a points and the slope this is the one you will primarily use in calculus
4:02so it says get the slope and a point and i can find the equation of he line so for example here i can use 10,3 or should i use 6,11, it doesnt matter ill use 10,3 so i get y minus 3 is the slope minus x minus 10. watch your minus sign now you are down, that is the equation of the line, you do not need to turn this
4:31into y equals negative 29x+23 you can feel free to do that, but you do not have to this is not better then this this form is how you can find the form of y-intercept 23, and this form you can do other stuff but the point slop form is just that simple you guys all seen this before i presume? this should be nice and easy
5:00because youre going to do a bunch of this in calc find the slope so lets practice one more time find the equation
5:42give me a point 2,-8 sounds good all you need to do is find the slope and plug in the formuls
6:04and what that tells us when you have slope remember .. the slope of -2 says anytime you go across 1, you go down 2 thats what the slope says if you look at this it says cross 1, down 2, cross 1, down 2 and i get there so here the slope is the difference in the ys. 5 minus -8
6:31the difference in the x and you get 13/4 its not the best choice of numbers but you know fractions ever since and then you pick, either points it does not matter so well do for 5 y minus 5 13/4 and x minus 6 all you need to know about a line and thats all were going to do with a line, find the slope
7:02and the line equation to tell you how to find the line yppu can draw it from there now were going to do a little more on conic section do you guys know what conic sections are? to test my art work here
7:31okay so imagine you had a cone and you have another cone and you draw them together like thar kind of like an hour glas if you then took a plane and you sliced straight across. parallel to face the shape you get is a circle
8:06if the slope of this plane was flat it has the same slope as the base when you slice across this cone you get a circle imagine you had a cone and you sliced it straight it will be a round thing a cone you slice, youll always get circles okay what is the tilt of the plane instead
8:36well youll still get something circler but itd be streched and thats ellipse so if the plane was no longer flat if it starts to angle then when you cut through the cone youll get something that will get elongated some how thats called ellipse if you then draawit where they are had you cut it
9:01thats why they are called conic secttions if you the slice it exactly parallel to this line you get something shaped like that and have a parabula its a picture of a horseshoe thats what the shape looks like and its a parabula
9:32if you slice it, tilt it more then this size of the column you will end up with shapes that look like that and those are called hurdles and these four things are conic sections so its not really, were not going to ask you to do it because the math is actually very complicated it will be important to you but not for me so conic sections
10:04are a circle elispse hyperbola parabula now technically a line a a point is conic section you get a line if you sorta just touch of the cones and youll get a point where you exactly went through the cones we dont care about those
10:31when we talk aout conic sections, we general gonna talk about these 4 what is a circle?
you see conic sections, you shouldve definitely see parabulas how ever you reword it you want too
11:01circles, have you guys heard of a hyperbola not so much. you did okay its not on the regentts okay good so you learned some new stuff arent these great pictures?
11:31if you look at your book theres really nice versions of these by professional artist imagine you were centered at the axis of a circle so what is a circle? a circle means that every point is the same distance from the center every point on the outside of the circle is always the same distance from the center of the circle we call this the radius
12:02and you got some point x,y and no matter where you go on the radius this x and this y will give you that radius and we have a relationship between x and y and r youll learn this is your favorite theorm try Pythagorean theorem, so we know x squared plus y squared will always be r squared and that is the equation of a circle
12:33the center of the origin okay remember an equation is just a relationship it just says i have a relationship between x and y, that if i square x and square y
13:04and you get the radius squred, Pythagorean theorem so you can do the x and the radius after you find y or i can find two possible ones because for example x is here it could be that x that x, we dont really know and thats why we the whole plus or mius thing for example
13:37if i know that x squared plus y squared is .5 then i know the radius of the circle is 5, because 5 squared is 25 this will be a circle center of the origin always draw the circle before you draw the axis, it makes it look more accurate draw this as=xis first and youll look more like this its the secret part to shooting first you shoot and then you boult
14:08it says if i have some random point x,y that distance is 5 so x squared plus y squared will only get 5 squared you know this, now.. what if i wanted to move that off of the origin if i wanted to translate it, if i want to shift it to the right or if i want to shift it up what i would do then
14:33i would do x-h squared plus y-k squared will equal r squared and that comes shifting it by h and k so this now has a center of h times k the radius is still r
15:30so by the equation of the line 5,11 radius of 4 and the equation is just x-5 squared plus y-11 squared equals 4 squared and you dont need to simplify that i mean you can but you dont need too but its not simpiliar if you multiply it by a half youll just get it in a different form if i multiplied this out i would get x squared minus 10x
16:04plus .5, plus y squared minus 22x y plus 121, equals 16 and that comes from multiplying out x minus 5, 1 is 11 and you sort of gather it together and i get x squared plus y squared minus 10x minus 22y plus
16:32130 equals 0 i could put it in that form but thats not better then this, in fact this is a pain in the neck to work with. this is great so this is really the form that you want sometimes you have to start with this form and put it in that form, remember how you do that you remeber how you do that?
completing the square aww did completing the square bring up nightmares?
17:04i have good news and bad news the bad news is im gonna do a completing the square and youre just going to have to follow along we dont really plan to test you on it unless youd like us too okay thats pretty much what youll need to know about circles
17:48this is the most important thing you will need to know from there we will probably ask you to do something like this
18:00alright im going to do a little erasing everybody feel good with this?
picture last one were gonna do parabulas now remember a parabula is also looks like a cone and parabulas are kind of fun to work with
18:33you all covered the basic one y equals x square does through the origin and to pick some x value 4, the y value will be 16. which is obviously not drawn to scale because 4 squared is 16 you have 1 and coma 1, 2 coma 4 and so on.
parabulas are very distinct for example in physics
19:02a foreign object possibly follows a shape of the parabula doesnt actually but the range you take is just to change in high school and projectiles and stuff alright so you use parabulas thats a goo approximation all of something closes to the earths surface can cause a parabula you throw it high enough gravity starts to be different so you dont really get a parabula
19:39its not exactly a parabula, when you throw a football, parabula, when you kick a soccerball parabula and so on so parabulas in general have a couple of things we want to know about first they can be upside down or right sode up second is they go through the x axis
20:01or not and the third is they have a vertex you can call the roots or the x intercepts or the zeros those three words can be changable they are called the x intercepts because they intercept the x axis they are called the zeros because you can find them by setting it equal to zero
20:34and they are the roots because solute to the equation if you use any of those three terms for the parabula now parabula coyuld be like upside down or right side up they could miss the x axis they could touch the x axis or it could go through the x axis you have ways to tell
21:02so this would have no zeros, not roots this would have one root and this will have two roots, no real roots so when you took the quadratic equation, to solve it you get an imaginary complex number for this one you get one root for this one, and you get two roots for that one so we will go through that for a little bit this is where we get the fun stuff i have a parabula and i want to find the vertex well..
21:34if its vertex form its easy but suppose i have the parabula y equals x squared+12x-20 so you can factor that, so you can say break out the two equations you guys remember how to do that?
22:03actually i think plus 20 works so this will factor into and that will tell me by setting this equal to zero that i get 0 at -10 0 at -2
22:32my parabula looks something like that and the vertex is half way between -10 and -2 another way to find this is by completing the square what completing the square means. Well depends that i want to find where it goes through 0
23:01so what i want to do is i want to write this as a perfect square so i take half of 12 and i square it first ill be finding the other side this gets easy and then the trick is to cut this in half
23:30the problem is this is x squared plus 12 x plus 36. so i have to add 6 squared to the other side or im gonna earse it again sorry so i took half of that number and squared it an added it to both sides and that becomes x plus 6 squares equals -20 plus 36
24:00you get x+6 squared, equals 16 take the square root you get x is -6 plus or minus 4 negative 6 plus 4 is -2, -6+4=-2, negative 6 minus 4 is 10 you never want the square root of something thats the technique, im gonna do it a couple more time because im gonna make sure you understand it
24:35what i am doing is i am taking my original x squared plus something else term and turing it into a perfect square and then i can set it equal to concert and take the square root so the technique is move the concept to the other side, cut the linear coefficent the b turned half, squared and add it to both sides
25:04lets do it again and make sure everyone gets it okay something like that and i want to find the 0's for x, if they are find by completing the squuare
25:31remember youre gonna have a test on this so follow along with it so first i put the 5 on the other side just to get it out of the way i cut the 8 in half and i square it and i added it to the left side which means i have to add it to the right side
26:01so far so good?
this now can be rewritten as x+4 squared because you take x plus 4 you square out and you get this and remember thats why i added the extra 4 squared so i can turn it into this but if i add 4 squared to the left side i have to add 4 squared to the right side it comes 11 now to solve it, i take plus or minus the square root
26:35and subtract 4 that helps me find the 0's. now i know where it crosses the x axis. yes?
i look at my recent two terms, i square plus 8x and when they trade and transform, where i can factor it into a perfect square
27:01and that form the way you get there is take the 8 and cut it in half which is 4 and squared and if i add it to the left side id have to add it to the right side so if i had x squared plus 10x minus 7 first i would add 7 to the other side
27:32where did the 8x go? when i factored this it became this because do it the other way, foil it out if you foil this out you get x plus 4 times x plus 4 is x squared plus 4x plus 4x plus 8x plus 16 so im taking this and turning it into a perfect square
28:00thats why its called completing the square, because you actually do a little geometric thing see it always works so i just want you guys to be comfortable with the algebra so that x square plus 10x minus 7 add that 7 to the other side and cut that 10 in half square it and if i add it to the left side i have to add it to the right side
28:30you see this before you look a little paste so this is now x plus 5 squared and this is 32 now youre gonna take plus or minus the square root and subtract 5 we will do one more of these, i want to make sure we are comfortable
29:02we will do something else here what if i had what if i had this one now there a 3x squared, thats annoying
29:31so first were gonna do exactly what we did before, add 5 to both sides and then divide everything by 3 okay cut the 4 in half and square it and add it to both sides
30:03so this becomes x plus 2 squared and that is 17/3 i do love fractions how did i go from 12 to 4? I divided everything by 3
30:30okay i had this annoying 3 from the 3x squared so i just divided through to get rid of it then i take half of this number square it and add it to both sides now i take the square roots and put a two on the other side and x=-2 plus or minus the square root
31:0017/3 which is pretty messy actually very messy we can make it worse so this is very annoying, these numbers and all this stuff but lets just suppose and say..yes?
whered i get 17/3 5/3 plus 4 is 17/3
31:31cause 4 is 12/5 i was hoping we would just go through that stuff real fast and wouldnt have to stop but i didnt quiet get away with that okay because we love fractions okay remeber all the steps i did?
so suppose instead of having all the numbers, i just had
32:01ax squared plus bx plus c equals 0 just to be a little algebraic so whats my first step? put the c on the other side and that was the first step and thats not so bad
32:33now i can divide everything bya and now for the annoying completing the square part i have to take b/a and cut it in half and have to square it that b over 2/a squared and i have to add it to both side
33:07oh this is getting really messy okay remember how this factors.. this now just factors to x plus b/2a sqaured and this is b squared thats 4a squared minus c/a
33:31so if i wanted to combined these i would need a common denominator so whats a common denominator from here and here?
4a squared so i multiply my top and bottom by 4a and now im gonna have x+(b/2a)squared and this is gonna be b squared/4a squared minus 4ac/4a squared it starts off familiar
34:04or o plus b over 2a, squared b squared minus 4ac 4a squared whats our next step?
our next step is take the square root and you get x plus b/2a equals plus or minus the square root of b squared minus 4ac
34:33and thats over 2a and now you put that on the other side and you get negative b plus or minus the square root of b squared minus 4ac over 2a you all wonder where that came from now you know isnt that exciting show grandma that at christmas
35:00say this is your tuition at work
35:44later and you can show this to you parents theres 7,000 dollars okay so now we did all that so what does that do?
it now tells you how to find the zeros where you cross the axis thats what the formula is for okay so of course you guys memorized that formula its one of the two things you memorized
36:05you know the song want to sing the song? no i always sang it but i sang 2 would be the moo so thats how you find the zeros for parabulas oh okay right no that was a sort of hip hop version of it, thats pretty good there
36:35you sure i dont mind. im not gonna erase it im gonna give you a few more minutes on the exam we are gonna as you to do that you should understand how to do this, fo a parabula you go through the x axis thats the two places its going through the x axis it tells us all sorts of funny things i wont erase that ill just cover it up for a second
37:24we are not going to ask you to complete the square we just want to see you understand whats going on and what you are doing
37:31its just a technique and frankly its just a way to take points off on the exam so that says if you have a parabula and it goes through the x axis theres two places were it cuts through it cuts through negative b plus the square root of b squared minus 4ac/2a and it cuts at negative b minus the square root of b squared minus 4ac over 2a
38:05pop goes the wiesel you sing it you get extra points the middle, well what are we doing?
we are taking negative b and adding it to square root of b squared minus 4ac and we are take negative b and subtracting b squared minus 4ac so half way between the two of them is just negative b/2a
38:36so thats the vertex the vertex is x=-b/2a so far so good?
alright you know some other remember you take -b and you add b squared minus 4ac and the you subtract the root of b squared minus 4ac okay but what if b squared minus 4ac was 0
39:02well then when we take b we add or subtract 0 there arent 2 points there is only 1 point so if b squared minus 4ac is zero, youre only gonna get 1 root, remember this is called the discriminate because the way it goes through the axis 1 time just over here because you move the end and you are subtracting anything
39:32what if b squared minus 4ac was a negative numbere?
well then you will be taking the square root of a negative number which is not a real number, its a complex number so you wouldnt be going through the axis at al so this would have no real roots and if b squared minus 4ac is greater then zero you will have 2 roots
40:24its also for the line segment its the axis of symmetry because parabulas are symmetric both go up the same amount on both sides
40:32so you can draw a line and youn say the left side is like the right side the equation of that line is x=-b/2a we happy so far?
so we could put this to work
41:15okay so say i had the parabula and i wanted to graph this parabula
41:33now i could factor it but now i have a handy formula, so why factor but i know a couple of things right away i know what the vertex of the parabula is the vertex is x=-b/2a which is positive 14 over 4 is 7/2
42:04so negative b is 14 2a is 4, 14 over 4 is 7/2 if i wanted to find the y coordinate of the vertex, i would now take 7/2 and i would plug in
42:35that gives you i think thats right, someone check that in a calculator is that right? 49/4 49/2 oh no thats not right 49/4
43:0049/2 minus 98/2 minus 49/2 i still -1/2 yup thats right will i see horrible numbers like this on the test? probably not okay it will be something easier
43:31its not a test of your ability to multiply fractions its an equation however this says the vertex is the point 7/2, -1/2 just matter what we did later now we want to find out where it crosses the x axis x=-b plus or minus the square roo b squared minus 4ac all over 2a
44:13thats thats 14 plus or minus 4 over 4 9/2 and
44:305/2 and the vertex is 7/2 thats half way between 5/2 and 9/2 good think i brought you thats 14 plus or minus 2 which is 16/4 which is 4 4 and 3 thats even better its still has the 7/2 in the middle but i like those numbers better
45:01by the way i made up one in my head okay so if i graph this cover that for a second
45:40by the way how do you find y intercept very easily you just put x=0 the y incept is 24 so that would make it 0 coma 24 how much are we loving parabulas?
not so much?
the main think you need to do with a parabula is you need to graph it
46:02if we give you the parabula so the easy way to graph it is to use the quadratic formula find the 2 zeros vertex is half way between those 2 zeros once you have the vertex and you have two other points you can graph the parabula can we erase and do it again?
dont erase, erase bad?
47:03so the easiest way to graph this, you cannot need to be factor you have to use quadratic firmula so x equals 16 plus or minus the square root b squared minus 4ac all over 2a
47:41no wonder why it wasnt working. thank you for who ever said that x is 16
48:09plus or minus the square root of 400 over 2 so 16 plus 20 is 36, over 2 is 18 16 minus 20 is minus 4
48:32over 2 is minus 2 that would something like that and now we have to find this thing by half way between minus 2 and 18 where does negative 2 come from?
i take 16 plus 20 divided by 2 i get 18 and 16 minus 20
49:01divided by 2 you get -2 so half way between -2 and 18, add them together you get 16 divide by 2 you get 8 the x coordinate of that is 8 and you plug the 8 in to get the y coordinate so lets see thats 64 minus 128 is minus 64 minus 36 is minus 100
49:33sounds good to me okay thats all that involved, if we can find this point thats thius the y intercept sure i get 8 and plug it back in the original equation so i get 8 squared minus 16 times 8
50:00minus 36 equals minus 100 when we go quickly through those means that i dont expect you to have to do that but you should always ask okay so we did lines, circles parabulas pretty good so wednesday we are gonna do ellipses and some other stuff