WEBVTT
Kind: captions
Language: en
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Okay so the first thing we are going to do is stuff about lines, you guys should be very comfortable with lines, right?
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when we graph, we draw a line to based on the relationship
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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?
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thats what we call slope.
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we 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?
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thats why you can use slope on line
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in other occurs the slope changes, thats what calculus is about, the one slope is changing
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and we can find the equation of the line if we know two different spots on the line
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so im assuming everybody can find the slope of a line with 3 points
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so lets practice
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can we use the one from webassign? sure
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this is something you shouldve done in AMP 103
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oh lets see
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so we have two points, so we know
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when we represent these, we say 10,3 and 6,11
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we say to ourself every time it goes down a certain amount, it goes across a certain amount of visa versa
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when you get to the slope you can really call it what you want
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so when you represent these algebraically, its gonna be
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y is the x to a single power, 1
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and well make an equation for you
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so whatn do we need? we need the slope
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slope as you all knw
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i hope
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is the change in y
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over the change in x
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or y2 minus y1 over x2 minus x1, it doesnt matter as long as its consistant
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what this means is
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for every certain change in y, i have a certain change in x
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and its always the same on a line
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as proposed to a parabola
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circle, fun graph
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and you can pick what ever points you want
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to be x1, y1 and x2,y2
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so how bout we make these
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number 1
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these number 2
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so this would be
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3-11
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and 10-6
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-8/4
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-2
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so far thats very straight forward
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and then we need an equation for a line
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so many of you love the equation mx+b
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but the truth is its better form of that equation that you want to use
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we potential get from rearranging this
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this is called the point slope form
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because it using a points
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and the slope
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this is the one you will primarily use in calculus
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so it says get the slope
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and a point and i can find the equation of he line
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so for example here
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i can use 10,3 or should i use 6,11, it doesnt matter
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ill use 10,3
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so i get y minus 3
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is the slope
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minus x minus 10. watch your minus sign
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now you are down, that is the equation of the line, you do not need to
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turn this
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into y equals negative 29x+23
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you can feel free to do that, but you do not have to
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this is not better then this
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this form is how you can find the form of y-intercept
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23, and this form you can do other stuff
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but the point slop form is just that simple
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you guys all seen this before i presume? this should be nice and easy
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because youre going to do a bunch of this in calc
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find the slope
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so lets practice one more time
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find the equation
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give me a point
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2,-8 sounds good
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all you need to do is find the slope and plug in the formuls
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and what that tells us
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when you have slope remember .. the slope of -2 says anytime you go across 1, you go down 2
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thats what the slope says
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if you look at this
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it says cross 1, down 2, cross 1, down 2
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and i get there
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so here the slope
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is the difference in the ys. 5 minus -8
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the difference in the x
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and you get 13/4
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its not the best choice of numbers but you know fractions ever since
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and then you pick, either points it does not matter
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so well do for 5
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y minus 5
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13/4
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and x minus 6
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all you need to know about a line
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and thats all were going to do with a line, find the slope
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and the line equation to tell you how to find the line
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yppu can draw it from there
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now were going to do a little more on conic section
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do you guys know what conic sections are? to test my art work here
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okay so imagine you had a cone
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and you have another cone and you draw them together like thar
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kind of like an hour glas
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if you then took a plane
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and you sliced straight across. parallel to face
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the shape you get is a circle
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if the slope of this plane was flat
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it has the same slope as the base
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when you slice across this cone you get a circle
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imagine you had a cone and you sliced it straight it will be a round thing
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a cone
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you slice, youll always get circles
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okay what is the tilt of the plane instead
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well youll still get something circler but itd be streched
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and thats ellipse
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so if the plane was no longer flat
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if it starts to angle
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then when you cut through the cone
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youll get something that will get elongated some how
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thats called ellipse
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if you then
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draawit where they are had you cut it
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thats why they are called conic secttions
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if you the slice it exactly parallel to this line
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you get something shaped like that and have a parabula
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its a picture of a horseshoe
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thats what the shape looks like and its a parabula
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if you slice it, tilt it more
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then
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this size of the column
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you will end up with shapes that look like that and those are called hurdles
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and these four things are conic sections
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so its not really, were not going to ask you to do it because the math is actually very complicated
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it will be important to you
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but not for me
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so conic sections
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are a circle
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elispse
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hyperbola
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parabula
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now technically a line a a point is conic section
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you get a line if you sorta just touch
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of the cones and youll get a point where you exactly went through the cones
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we dont care about those
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when we talk aout conic sections, we general gonna talk about these 4
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what is a circle?
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you see conic sections, you shouldve definitely see parabulas
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how ever you reword it you want too
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circles, have you guys heard of a hyperbola
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not so much. you did okay
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its not on the regentts
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okay good so you learned some new stuff
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arent these great pictures?
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if you look at your book theres really nice versions of these
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by professional artist
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imagine you were centered at the axis of a circle
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so what is a circle? a circle means
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that every point is the same distance from the center
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every point on the outside of the circle is always the same distance from the center of the circle
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we call this the radius
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and you got some point x,y
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and no matter where you go on the radius
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this x and this y will give you that radius
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and we have a relationship between x and y and r
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youll learn this is your favorite theorm
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try Pythagorean theorem, so we know x squared plus y squared will always be r squared and that is the equation of a circle
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the center of the origin
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okay remember an equation is just a relationship
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it just says
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i have a relationship between x and y, that if i square x and square y
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and you get the radius squred, Pythagorean theorem
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so you can do the x and the radius after you find y
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or i can find two possible ones
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because for example x is here
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it could be that x that x, we dont really know and thats why we the whole plus or mius thing
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for example
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if i know that x squared plus y squared is .5
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then i know the radius of the circle is 5, because 5 squared is 25
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this will be a circle
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center of the origin always draw the circle before you draw the axis, it makes it look more accurate
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draw this as=xis first and youll look more like this
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its the secret part to shooting first you shoot and then you boult
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it says if i have some random point x,y
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that distance is 5 so x squared plus y squared will only get 5 squared
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you know this, now.. what if i wanted to move that off of the origin
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if i wanted to translate it, if i want to shift it to the right or if i want to shift it up
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what i would do then
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i would do x-h squared
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plus y-k squared
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will equal r squared
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and that comes shifting it by h and k
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so this now has a center of h times k
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the radius is still r
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so by the equation of the line 5,11
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radius of 4
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and the equation is just x-5 squared
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plus y-11 squared
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equals 4 squared and you dont need to simplify that
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i mean you can but you dont need too
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but its not simpiliar if you multiply it by a half
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youll just get it in a different form
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if i multiplied this out i would get x squared minus 10x
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plus .5, plus y squared minus 22x
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y plus 121, equals 16 and that comes from multiplying out
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x minus 5, 1 is 11
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and you sort of gather it together and i get x squared
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plus y squared
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minus 10x minus 22y
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plus
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130 equals 0
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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
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so this is really the form that you want
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sometimes you have to start with this form and put it in that form, remember how you do that
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you remeber how you do that?
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completing the square
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aww did completing the square bring up nightmares?
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i have good news and bad news
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the bad news is im gonna do a completing the square and youre just going to have to follow along
00:17:14.600 --> 00:17:16.920
we dont really plan to test you on it
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unless youd like us too
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okay thats pretty much what youll need to know about circles
00:17:48.900 --> 00:17:50.740
this is the most important thing you will need to know
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from there we will probably ask you to do something like this
00:18:00.740 --> 00:18:02.740
alright im going to do a little erasing
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everybody feel good with this?
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picture last one
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were gonna do parabulas
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now remember a parabula is also looks like a cone
00:18:29.040 --> 00:18:31.840
and parabulas are kind of fun to work with
00:18:33.120 --> 00:18:35.040
you all covered the basic one
00:18:39.900 --> 00:18:42.620
y equals x square does through the origin
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and to pick some x value 4, the y value will be 16. which is obviously not drawn to scale
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because 4 squared is 16
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you have 1 and coma 1, 2 coma 4 and so on.
00:18:59.360 --> 00:19:02.380
parabulas are very distinct for example in physics
00:19:02.760 --> 00:19:07.180
a foreign object possibly follows a shape of the parabula
00:19:07.260 --> 00:19:08.380
doesnt actually
00:19:08.920 --> 00:19:11.480
but the range you take is just to change
00:19:11.480 --> 00:19:16.880
in high school and projectiles and stuff alright so you use parabulas thats a goo approximation
00:19:16.880 --> 00:19:20.040
all of something closes to the earths surface can cause a parabula
00:19:20.040 --> 00:19:23.240
you throw it high enough gravity starts to be different
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so you dont really get a parabula
00:19:39.240 --> 00:19:46.240
its not exactly a parabula, when you throw a football, parabula, when you kick a soccerball parabula and so on
00:19:47.000 --> 00:19:51.960
so parabulas in general have a couple of things we want to know about
00:19:54.120 --> 00:19:57.080
first they can be upside down or right sode up
00:19:58.180 --> 00:20:00.500
second is they go through the x axis
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or not
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and the third is they have a vertex
00:20:08.940 --> 00:20:12.840
you can call the roots or the x intercepts
00:20:16.920 --> 00:20:17.720
or the zeros
00:20:23.180 --> 00:20:25.500
those three words can be changable
00:20:25.500 --> 00:20:28.520
they are called the x intercepts because they intercept the x axis
00:20:28.540 --> 00:20:33.660
they are called the zeros because you can find them by setting it equal to zero
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and they are the roots because solute to the equation
00:20:37.300 --> 00:20:40.660
if you use any of those three terms for the parabula
00:20:41.280 --> 00:20:42.640
now parabula coyuld
00:20:44.280 --> 00:20:46.600
be like upside down or right side up
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they could miss the x axis
00:20:53.240 --> 00:20:55.720
they could touch the x axis
00:20:56.460 --> 00:20:58.540
or it could go through the x axis
00:20:59.660 --> 00:21:01.020
you have ways to tell
00:21:02.220 --> 00:21:05.160
so this would have no zeros, not roots
00:21:05.160 --> 00:21:08.600
this would have one root and this will have two roots, no real roots
00:21:10.340 --> 00:21:16.180
so when you took the quadratic equation, to solve it you get an imaginary complex number for this one
00:21:16.380 --> 00:21:20.460
you get one root for this one, and you get two roots for that one
00:21:22.180 --> 00:21:23.620
so we will go through that for a little bit
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this is where we get the fun stuff
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i have a parabula and i want to find the vertex well..
00:21:34.940 --> 00:21:51.140
if its vertex form its easy but suppose i have the parabula y equals x squared+12x-20
00:21:54.220 --> 00:21:58.460
so you can factor that, so you can say break out the two equations
00:21:59.160 --> 00:22:01.320
you guys remember how to do that?
00:22:03.600 --> 00:22:04.240
actually
00:22:06.120 --> 00:22:07.480
i think plus 20 works
00:22:09.700 --> 00:22:11.740
so this will factor into
00:22:17.920 --> 00:22:19.280
and that will tell me
00:22:21.080 --> 00:22:23.000
by setting this equal to zero
00:22:24.100 --> 00:22:26.000
that i get 0 at -10
00:22:27.540 --> 00:22:29.340
0 at -2
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my parabula looks something like that
00:22:34.460 --> 00:22:37.680
and the vertex is half way between -10 and -2
00:22:50.460 --> 00:22:52.940
another way to find this is by completing the square
00:22:53.040 --> 00:23:00.300
what completing the square means. Well depends that i want to find where it goes through 0
00:23:01.800 --> 00:23:05.720
so what i want to do is i want to write this as a perfect square
00:23:06.960 --> 00:23:09.740
so i take half of 12
00:23:10.580 --> 00:23:11.540
and i square it
00:23:16.740 --> 00:23:19.060
first ill be finding the other side
00:23:20.700 --> 00:23:21.660
this gets easy
00:23:23.140 --> 00:23:24.900
and then the trick is to cut this in half
00:23:30.160 --> 00:23:32.800
the problem is this is x squared plus 12 x
00:23:34.500 --> 00:23:38.880
plus 36. so i have to add 6 squared to the other side
00:23:39.420 --> 00:23:41.500
or im gonna earse it again sorry
00:23:50.520 --> 00:23:54.680
so i took half of that number and squared it an added it to both sides and that becomes
00:23:55.040 --> 00:23:58.820
x plus 6 squares equals -20 plus 36
00:24:00.680 --> 00:24:05.760
you get x+6 squared, equals 16
00:24:06.860 --> 00:24:08.220
take the square root
00:24:14.260 --> 00:24:17.440
you get x is -6 plus or minus 4
00:24:18.240 --> 00:24:23.020
negative 6 plus 4 is -2, -6+4=-2, negative 6 minus 4 is 10
00:24:23.660 --> 00:24:26.540
you never want the square root of something
00:24:26.760 --> 00:24:33.240
thats the technique, im gonna do it a couple more time because im gonna make sure you understand it
00:24:35.460 --> 00:24:44.720
what i am doing is i am taking my original x squared plus something else term and turing it into a perfect square
00:24:45.160 --> 00:24:49.160
and then i can set it equal to concert and take the square root
00:24:50.360 --> 00:25:00.660
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
00:25:04.360 --> 00:25:05.880
lets do it again and make sure everyone gets it
00:25:22.440 --> 00:25:24.120
okay something like that
00:25:25.140 --> 00:25:29.940
and i want to find the 0's for x, if they are find by completing the squuare
00:25:31.040 --> 00:25:33.600
remember youre gonna have a test on this so follow along with it
00:25:33.620 --> 00:25:35.940
so first i put the 5 on the other side
00:25:40.680 --> 00:25:42.440
just to get it out of the way
00:25:47.840 --> 00:25:50.540
i cut the 8 in half and i square it
00:25:53.980 --> 00:25:57.720
and i added it to the left side which means i have to add it to the right side
00:26:01.900 --> 00:26:02.860
so far so good?
00:26:04.760 --> 00:26:05.320
this now
00:26:06.280 --> 00:26:10.560
can be rewritten as x+4 squared
00:26:11.260 --> 00:26:14.860
because you take x plus 4 you square out and you get this
00:26:16.780 --> 00:26:21.660
and remember thats why i added the extra 4 squared so i can turn it into this
00:26:21.660 --> 00:26:25.240
but if i add 4 squared to the left side i have to add 4 squared to the right side
00:26:25.800 --> 00:26:26.880
it comes 11
00:26:28.460 --> 00:26:32.040
now to solve it, i take plus or minus the square root
00:26:35.180 --> 00:26:36.140
and subtract 4
00:26:44.420 --> 00:26:48.980
that helps me find the 0's. now i know where it crosses the x axis. yes?
00:26:52.660 --> 00:26:55.620
i look at my recent two terms, i square plus 8x
00:26:55.620 --> 00:27:00.540
and when they trade and transform, where i can factor it into a perfect square
00:27:01.020 --> 00:27:09.400
and 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
00:27:09.720 --> 00:27:17.540
so if i had x squared plus 10x minus 7
00:27:19.980 --> 00:27:21.200
first i would
00:27:22.200 --> 00:27:23.640
add 7 to the other side
00:27:32.900 --> 00:27:36.500
where did the 8x go? when i factored this it became this
00:27:40.380 --> 00:27:42.040
because do it the other way, foil it out
00:27:42.320 --> 00:27:44.160
if you foil this out you get
00:27:44.980 --> 00:27:48.380
x plus 4 times x plus 4
00:27:49.260 --> 00:27:55.360
is x squared plus 4x plus 4x plus 8x plus 16
00:27:56.080 --> 00:28:00.580
so im taking this and turning it into a perfect square
00:28:00.580 --> 00:28:05.740
thats why its called completing the square, because you actually do a little geometric thing
00:28:06.200 --> 00:28:07.480
see it always works
00:28:09.220 --> 00:28:11.960
so i just want you guys to be comfortable with the algebra
00:28:12.880 --> 00:28:14.960
so that x square plus 10x minus 7
00:28:15.360 --> 00:28:18.560
add that 7 to the other side and cut that 10 in half
00:28:21.380 --> 00:28:22.020
square it
00:28:22.380 --> 00:28:24.300
and if i add it to the left side
00:28:24.820 --> 00:28:26.900
i have to add it to the right side
00:28:30.420 --> 00:28:33.220
you see this before you look a little paste
00:28:34.440 --> 00:28:42.400
so this is now x plus 5 squared and this is 32
00:28:42.800 --> 00:28:46.080
now youre gonna take plus or minus the square root
00:28:50.900 --> 00:28:51.860
and subtract 5
00:28:57.160 --> 00:29:01.480
we will do one more of these, i want to make sure we are comfortable
00:29:02.500 --> 00:29:04.100
we will do something else here
00:29:11.040 --> 00:29:11.840
what if i had
00:29:23.020 --> 00:29:24.380
what if i had this one
00:29:26.760 --> 00:29:29.320
now there a 3x squared, thats annoying
00:29:31.100 --> 00:29:33.700
so first were gonna do exactly what we did before, add 5 to both sides
00:29:38.540 --> 00:29:40.620
and then divide everything by 3
00:29:52.480 --> 00:29:55.320
okay cut the 4 in half and square it
00:29:56.080 --> 00:29:57.600
and add it to both sides
00:30:03.900 --> 00:30:08.380
so this becomes x plus 2 squared
00:30:09.040 --> 00:30:12.100
and that is 17/3
00:30:12.900 --> 00:30:14.180
i do love fractions
00:30:26.980 --> 00:30:29.460
how did i go from 12 to 4? I divided everything by 3
00:30:30.500 --> 00:30:36.100
okay i had this annoying 3 from the 3x squared so i just divided through to get rid of it
00:30:36.900 --> 00:30:38.900
then i take half of this number
00:30:39.260 --> 00:30:41.420
square it and add it to both sides
00:30:44.960 --> 00:30:46.720
now i take the square roots
00:30:51.960 --> 00:30:53.880
and put a two on the other side
00:30:54.880 --> 00:30:57.280
and x=-2
00:30:57.740 --> 00:30:59.660
plus or minus the square root
00:31:00.840 --> 00:31:01.780
17/3
00:31:02.700 --> 00:31:04.140
which is pretty messy
00:31:08.880 --> 00:31:10.240
actually very messy
00:31:10.720 --> 00:31:12.000
we can make it worse
00:31:16.940 --> 00:31:19.840
so this is very annoying, these numbers and all this stuff
00:31:19.840 --> 00:31:22.200
but lets just suppose and say..yes?
00:31:25.600 --> 00:31:26.940
whered i get 17/3
00:31:27.940 --> 00:31:30.760
5/3 plus 4 is 17/3
00:31:31.600 --> 00:31:32.560
cause 4 is 12/5
00:31:37.040 --> 00:31:40.020
i was hoping we would just go through that stuff real fast and wouldnt have to stop
00:31:40.940 --> 00:31:43.180
but i didnt quiet get away with that
00:31:44.160 --> 00:31:46.240
okay because we love fractions
00:31:55.220 --> 00:31:57.380
okay remeber all the steps i did?
00:31:57.380 --> 00:32:00.680
so suppose instead of having all the numbers, i just had
00:32:01.120 --> 00:32:01.840
ax squared
00:32:03.120 --> 00:32:06.440
plus bx plus c equals 0
00:32:09.800 --> 00:32:11.640
just to be a little algebraic
00:32:14.580 --> 00:32:17.780
so whats my first step? put the c on the other side
00:32:28.980 --> 00:32:30.360
and that was the first step and thats not so bad
00:32:33.060 --> 00:32:35.140
now i can divide everything bya
00:32:41.780 --> 00:32:45.220
and now for the annoying completing the square part
00:32:46.100 --> 00:32:48.600
i have to take b/a and cut it in half
00:32:49.660 --> 00:32:51.020
and have to square it
00:32:54.620 --> 00:32:57.080
that b over 2/a
00:32:58.080 --> 00:33:01.540
squared and i have to add it to both side
00:33:07.620 --> 00:33:09.700
oh this is getting really messy
00:33:12.460 --> 00:33:16.380
okay remember how this factors.. this now just factors to x
00:33:16.620 --> 00:33:18.380
plus b/2a
00:33:19.780 --> 00:33:20.340
sqaured
00:33:21.260 --> 00:33:23.280
and this is b squared
00:33:24.320 --> 00:33:26.440
thats 4a squared
00:33:27.920 --> 00:33:28.560
minus c/a
00:33:31.580 --> 00:33:33.660
so if i wanted to combined these
00:33:33.660 --> 00:33:35.440
i would need a common denominator
00:33:35.440 --> 00:33:38.260
so whats a common denominator from here and here?
00:33:38.260 --> 00:33:39.600
4a squared
00:33:40.480 --> 00:33:42.800
so i multiply my top and bottom by 4a
00:33:43.640 --> 00:33:45.000
and now im gonna have
00:33:45.840 --> 00:33:50.160
x+(b/2a)squared
00:33:50.160 --> 00:33:53.640
and this is gonna be b squared/4a squared
00:33:53.900 --> 00:33:58.660
minus 4ac/4a squared
00:33:59.300 --> 00:34:00.820
it starts off familiar
00:34:04.320 --> 00:34:04.820
or
00:34:05.920 --> 00:34:08.680
o plus b over 2a, squared
00:34:09.980 --> 00:34:11.820
b squared minus 4ac
00:34:12.840 --> 00:34:14.400
4a squared
00:34:16.520 --> 00:34:17.880
whats our next step?
00:34:21.680 --> 00:34:24.080
our next step is take the square root
00:34:24.560 --> 00:34:27.000
and you get x plus b/2a
00:34:27.660 --> 00:34:32.860
equals plus or minus the square root of b squared minus 4ac
00:34:33.440 --> 00:34:35.140
and thats over 2a
00:34:36.380 --> 00:34:38.780
and now you put that on the other side
00:34:39.940 --> 00:34:47.300
and you get negative b plus or minus the square root of b squared minus 4ac over 2a
00:34:47.300 --> 00:34:48.920
you all wonder where that came from
00:34:49.160 --> 00:34:49.960
now you know
00:34:56.020 --> 00:34:57.300
isnt that exciting
00:34:58.040 --> 00:35:00.120
show grandma that at christmas
00:35:00.580 --> 00:35:02.660
say this is your tuition at work
00:35:44.540 --> 00:35:48.700
later and you can show this to you parents theres 7,000 dollars
00:35:49.580 --> 00:35:52.980
okay so now we did all that so what does that do?
00:35:53.260 --> 00:35:55.660
it now tells you how to find the zeros
00:35:56.060 --> 00:35:57.660
where you cross the axis
00:35:58.040 --> 00:35:59.960
thats what the formula is for
00:35:59.960 --> 00:36:04.920
okay so of course you guys memorized that formula its one of the two things you memorized
00:36:05.300 --> 00:36:06.420
you know the song
00:36:07.980 --> 00:36:09.580
want to sing the song? no
00:36:10.760 --> 00:36:13.860
i always sang it but i sang 2 would be the moo
00:36:17.540 --> 00:36:20.500
so thats how you find the zeros for parabulas
00:36:21.160 --> 00:36:22.040
oh okay right
00:36:27.460 --> 00:36:31.780
no that was a sort of hip hop version of it, thats pretty good there
00:36:35.820 --> 00:36:40.860
you sure i dont mind. im not gonna erase it im gonna give you a few more minutes
00:36:42.360 --> 00:36:44.920
on the exam we are gonna as you to do that
00:36:47.120 --> 00:36:51.580
you should understand how to do this, fo a parabula
00:36:51.660 --> 00:36:53.260
you go through the x axis
00:36:53.260 --> 00:36:56.320
thats the two places its going through the x axis
00:36:56.560 --> 00:36:58.960
it tells us all sorts of funny things
00:36:58.960 --> 00:37:01.520
i wont erase that ill just cover it up for a second
00:37:24.380 --> 00:37:26.720
we are not going to ask you to complete the square
00:37:26.720 --> 00:37:29.540
we just want to see you understand whats going on and what you are doing
00:37:31.000 --> 00:37:33.340
its just a technique and frankly
00:37:33.620 --> 00:37:36.420
its just a way to take points off on the exam
00:37:37.840 --> 00:37:40.080
so that says if you have a parabula
00:37:43.120 --> 00:37:45.040
and it goes through the x axis
00:37:46.000 --> 00:37:48.560
theres two places were it cuts through
00:37:49.140 --> 00:37:51.600
it cuts through negative b plus
00:37:51.660 --> 00:37:54.380
the square root of b squared minus 4ac/2a
00:37:56.400 --> 00:38:04.640
and it cuts at negative b minus the square root of b squared minus 4ac over 2a
00:38:05.500 --> 00:38:06.780
pop goes the wiesel
00:38:13.140 --> 00:38:15.220
you sing it you get extra points
00:38:17.900 --> 00:38:20.660
the middle, well what are we doing?
00:38:20.660 --> 00:38:28.460
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
00:38:28.520 --> 00:38:32.200
so half way between the two of them is just negative b/2a
00:38:36.040 --> 00:38:37.320
so thats the vertex
00:38:41.060 --> 00:38:43.600
the vertex is x=-b/2a
00:38:45.780 --> 00:38:46.740
so far so good?
00:38:47.960 --> 00:38:49.800
alright you know some other
00:38:50.880 --> 00:38:54.240
remember you take -b and you add b squared minus 4ac
00:38:54.240 --> 00:38:59.000
and the you subtract the root of b squared minus 4ac
00:38:59.000 --> 00:39:01.380
okay but what if b squared minus 4ac was 0
00:39:02.980 --> 00:39:05.780
well then when we take b we add or subtract 0
00:39:05.940 --> 00:39:08.660
there arent 2 points there is only 1 point
00:39:09.240 --> 00:39:12.880
so if b squared minus 4ac
00:39:12.980 --> 00:39:14.900
is zero, youre only gonna get
00:39:14.900 --> 00:39:18.180
1 root, remember this is called the discriminate
00:39:21.520 --> 00:39:24.560
because the way it goes through the axis 1 time
00:39:24.560 --> 00:39:25.520
just over here
00:39:27.780 --> 00:39:30.800
because you move the end and you are subtracting anything
00:39:32.240 --> 00:39:35.600
what if b squared minus 4ac was a negative numbere?
00:39:37.800 --> 00:39:42.900
well then you will be taking the square root of a negative number which is not a real number, its a complex number
00:39:42.900 --> 00:39:45.460
so you wouldnt be going through the axis at al
00:39:45.940 --> 00:39:49.620
so this would have no real roots
00:39:52.520 --> 00:39:56.920
and if b squared minus 4ac is greater then zero you will have 2 roots
00:40:24.620 --> 00:40:26.540
its also for the line segment
00:40:26.540 --> 00:40:31.980
its the axis of symmetry because parabulas are symmetric both go up the same amount on both sides
00:40:32.200 --> 00:40:35.160
so you can draw a line
00:40:36.260 --> 00:40:39.380
and youn say the left side is like the right side
00:40:39.980 --> 00:40:44.080
the equation of that line is x=-b/2a
00:40:48.280 --> 00:40:49.320
we happy so far?
00:40:52.400 --> 00:40:54.160
so we could put this to work
00:41:15.280 --> 00:41:17.200
okay so say i had the parabula
00:41:26.140 --> 00:41:28.460
and i wanted to graph this parabula
00:41:33.400 --> 00:41:35.140
now i could factor it
00:41:35.160 --> 00:41:38.040
but now i have a handy formula, so why factor
00:41:39.000 --> 00:41:41.560
but i know a couple of things right away
00:41:41.560 --> 00:41:43.920
i know what the vertex of the parabula is
00:41:44.420 --> 00:41:45.300
the vertex is
00:41:49.540 --> 00:41:52.020
x=-b/2a
00:41:53.960 --> 00:41:55.320
which is positive 14
00:41:55.840 --> 00:41:56.660
over 4
00:41:57.540 --> 00:41:58.040
is 7/2
00:42:04.840 --> 00:42:06.040
so negative b is 14
00:42:07.080 --> 00:42:09.280
2a is 4, 14 over 4 is 7/2
00:42:12.760 --> 00:42:17.320
if i wanted to find the y coordinate of the vertex, i would now take 7/2
00:42:18.080 --> 00:42:19.280
and i would plug in
00:42:35.980 --> 00:42:36.940
that gives you
00:42:39.520 --> 00:42:43.200
i think thats right, someone check that in a calculator
00:42:46.780 --> 00:42:48.060
is that right? 49/4
00:42:48.300 --> 00:42:49.900
49/2
00:42:50.640 --> 00:42:52.000
oh no thats not right
00:42:55.820 --> 00:42:56.320
49/4
00:43:00.980 --> 00:43:01.480
49/2
00:43:02.160 --> 00:43:02.880
minus 98/2
00:43:04.840 --> 00:43:05.820
minus 49/2
00:43:07.020 --> 00:43:07.820
i still -1/2
00:43:22.200 --> 00:43:23.240
yup thats right
00:43:25.460 --> 00:43:28.200
will i see horrible numbers like this on the test? probably not
00:43:28.820 --> 00:43:30.980
okay it will be something easier
00:43:31.860 --> 00:43:35.300
its not a test of your ability to multiply fractions
00:43:35.900 --> 00:43:36.440
its an equation
00:43:36.440 --> 00:43:36.940
however
00:43:36.940 --> 00:43:42.940
this says the vertex is the point 7/2, -1/2
00:43:42.940 --> 00:43:44.680
just matter what we did later
00:43:44.680 --> 00:43:46.900
now we want to find out where it crosses the x axis
00:43:49.020 --> 00:43:50.480
x=-b
00:43:51.020 --> 00:43:57.660
plus or minus the square roo b squared minus 4ac
00:43:59.420 --> 00:44:00.140
all over 2a
00:44:13.580 --> 00:44:14.080
thats
00:44:20.340 --> 00:44:24.700
thats 14 plus or minus 4 over 4
00:44:26.280 --> 00:44:26.780
9/2
00:44:28.580 --> 00:44:29.080
and
00:44:30.060 --> 00:44:30.560
5/2
00:44:32.100 --> 00:44:35.700
and the vertex is 7/2 thats half way between 5/2 and 9/2
00:44:47.900 --> 00:44:49.500
good think i brought you
00:44:49.560 --> 00:44:51.080
thats 14 plus or minus 2
00:44:52.960 --> 00:44:54.780
which is 16/4 which is 4
00:44:55.020 --> 00:44:56.620
4 and 3 thats even better
00:44:56.620 --> 00:44:58.180
its still has the 7/2 in the middle
00:44:58.560 --> 00:45:00.640
but i like those numbers better
00:45:01.620 --> 00:45:03.700
by the way i made up one in my head
00:45:05.160 --> 00:45:06.600
okay so if i graph this
00:45:13.420 --> 00:45:14.940
cover that for a second
00:45:40.940 --> 00:45:43.300
by the way how do you find y intercept very easily
00:45:43.420 --> 00:45:45.660
you just put x=0
00:45:46.160 --> 00:45:49.600
the y incept is 24 so that would make it 0 coma 24
00:45:53.360 --> 00:45:55.600
how much are we loving parabulas?
00:45:57.060 --> 00:45:57.860
not so much?
00:45:57.860 --> 00:46:01.680
the main think you need to do with a parabula is you need to graph it
00:46:02.040 --> 00:46:03.800
if we give you the parabula
00:46:04.520 --> 00:46:08.280
so the easy way to graph it is to use the quadratic formula
00:46:08.920 --> 00:46:09.960
find the 2 zeros
00:46:10.580 --> 00:46:13.220
vertex is half way between those 2 zeros
00:46:13.220 --> 00:46:17.140
once you have the vertex and you have two other points you can graph the parabula
00:46:20.380 --> 00:46:22.220
can we erase and do it again?
00:46:24.920 --> 00:46:26.200
dont erase, erase bad?
00:47:03.800 --> 00:47:10.040
so the easiest way to graph this, you cannot need to be factor you have to use quadratic firmula
00:47:11.200 --> 00:47:13.100
so x equals 16
00:47:13.780 --> 00:47:15.700
plus or minus the square root
00:47:20.540 --> 00:47:22.140
b squared minus 4ac
00:47:23.280 --> 00:47:24.520
all over 2a
00:47:41.220 --> 00:47:44.460
no wonder why it wasnt working. thank you for who ever said that
00:47:59.940 --> 00:48:00.440
x is 16
00:48:09.580 --> 00:48:13.500
plus or minus the square root of 400 over 2
00:48:24.200 --> 00:48:28.300
so 16 plus 20 is 36, over 2 is 18
00:48:29.600 --> 00:48:31.440
16 minus 20 is minus 4
00:48:32.020 --> 00:48:33.660
over 2 is minus 2
00:48:37.860 --> 00:48:42.100
that would something like that and now we have to find this thing
00:48:42.920 --> 00:48:48.680
by half way between minus 2 and 18
00:48:51.040 --> 00:48:53.200
where does negative 2 come from?
00:48:54.600 --> 00:48:57.100
i take 16 plus 20
00:48:57.420 --> 00:49:01.560
divided by 2 i get 18 and 16 minus 20
00:49:01.660 --> 00:49:04.160
divided by 2 you get -2
00:49:08.740 --> 00:49:13.780
so half way between -2 and 18, add them together you get 16 divide by 2 you get 8
00:49:15.180 --> 00:49:18.300
the x coordinate of that is 8 and you plug the 8 in
00:49:18.820 --> 00:49:20.340
to get the y coordinate
00:49:22.100 --> 00:49:23.900
so lets see thats 64
00:49:24.560 --> 00:49:26.560
minus 128 is minus 64
00:49:26.560 --> 00:49:29.780
minus 36 is minus 100
00:49:33.500 --> 00:49:34.620
sounds good to me
00:49:38.440 --> 00:49:42.040
okay thats all that involved, if we can find this point
00:49:45.540 --> 00:49:46.340
thats thius
00:49:46.940 --> 00:49:47.760
the y intercept
00:49:50.980 --> 00:49:53.360
sure i get 8 and plug it back in the original equation
00:49:54.120 --> 00:49:55.540
so i get
00:49:56.940 --> 00:49:57.580
8 squared
00:49:58.240 --> 00:49:59.960
minus 16 times 8
00:50:00.820 --> 00:50:01.380
minus 36
00:50:02.460 --> 00:50:03.580
equals minus 100
00:50:09.060 --> 00:50:12.580
when we go quickly through those means that i dont expect you to have to do that
00:50:13.560 --> 00:50:15.240
but you should always ask
00:50:17.060 --> 00:50:20.200
okay so we did lines, circles parabulas
00:50:21.260 --> 00:50:21.760
pretty good
00:50:22.900 --> 00:50:26.400
so wednesday we are gonna do ellipses and some other stuff