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this video you’ll learn how to plot points in the cartesian plane the right way

but did you know the discoverer of the coordinate plane liked to stay in bed

late and then he started watching it fly on the wall one day

and rather than get up and start his day he began to model the fly’s movements

relative to a corner of the walls thus began analytic geometry

so who was this procrastinator let’s find out hi everyone welcome back i’m dave

in this video how to plot points in the cartesian plane the right way

is part of the series functions and their graphs step-by-step tutorials for

pre-calculus the link to the series is below in the

description i hope that you check it out and so let’s see what’s up for today so

today we’re going to uh talk about some background information of the cartesian

plane then we’re going to plot some points uh

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we’re talking about the cartesian planes are going to be plotting points in 2d

and we’re going to talk uh about what the cartesian plane is and then we’ll do

some um fun examples where we’re finding the coordinates of a point and then

we’ll you know given some information determine the quadrants of the point so

let’s go ahead and get started so first is a little bit of background

information about the cartesian plane so the cartesian plane is fundamental in

the sense that the development of the cartesian coordinate system

will play a fundamental role in the development of the calculus uh which

was discovered by newton and leibniz and so the two coordinate description of

the plane was later generalized into the concept of vector spaces

so you know the cartesian plane wasn’t just

so solely for this discovery but it has led to a lot of great advancements in

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mathematics so the discovery itself um refers to the

french mathematician renee descartes who published this idea in 1637

um it was also independently discovered by vermont who also worked in three

three dimensions but he didn’t publish uh his three dimension discovery here

okay and so um what we’re going to do up first is um plotting some points in 2d

so let’s go ahead and look at that so when you want to plot some points

here you want to make sure and have a nice straight edge so i’m going to use my

track pad here to maybe get some edges here i can turn my board sideways here

there we go so i have some nice coordinate axes there

and i’m going to label this the x-axis and this is the y-axis

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and now i’m going to lay down some tick marks so here we go so there’s one two

three four five one two three four five one two three four five it may of course

help if you have graphing paper where these tick marks are already laid out

but in any case we’re going to plot some points so let’s plot the point here 2 3

so we’re going to come over here 2 and then we’re going to come up here 3 1

2 3 so that’s about right here so this is the point here two three

this point right here is zero zero that’s called the origin

and this would be the point right here minus three and then we’ll go down here

to say minus five so that would be the point here minus three minus five

so in general we have the uh x and y point

and this is called the x coordinate and this is called the y coordinate and this

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is an ordered pair as opposed to an open interval we would

use the same notation for open interval but in this context we’re using this

notation right here to represent an ordered pair the first number is the

distance from the origin the sign distance from the origin and

then y is the sign distance from the origin along the y axis here along the

vertical axis and so let’s just plot one more point

let’s say we plot the point here one two three four five

um in fact let’s go minus five one two three four five here’s minus five and

then here’s positive five so this would be the point here minus five five

right there one two three four five and one two three four five oops

okay so there’s a couple of points there that we plotted

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and i think we can get the idea from there about plotting these points here

okay so up next here let’s look at exactly what is the cartesian plane

i’m going to look at a little bit more background here first

so first off the invention itself was very crucial

so the invention came in 17th century as i said before rene descartes

he revolutionized my mathematics by giving the first systematic link between

euclidean geometry which had been known for thousands of years

and tying it together with algebra using variables

and so this is foundational in the sense that there’s a whole lot of other

subjects that have benefited from from the cartesian coordinates

foundation for analytic geometry in many other branches of mathematics such as

linear now uh linear algebra complex analysis

so you’re not going to just study in pre-calculus it’s going to be

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fundamental for many many many topics many subjects and its applications are

found throughout the sciences so i’ll just mention a couple astronomy

physics engineering the most common coordinate system used in

computer graphics computer aided geometric design and other geometry data

processing so the cartesian plane is truly

uh foundational and applied throughout the sciences

and um so the cartesian plane is formed by using two real number lines

so we need the real number lines is what we’re going to use in pre-calculus

they’re going to be intersecting at right angles so we often don’t

label these right angles but they’re there so let’s go ahead and

draw another coordinate axes here i’ll use my trackpad again here

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to get me a nice solid line here and so let’s just chop this off here

in here and this is the uh horizontal axis and i want to label x

vertical axis of and so what i’m saying right here is uh intersecting right

angles this is a right angle here but usually it’s not even bothered to be to

be labeled um all of these are right angles here all four of them and so

you know we’re going to have the um the x-axis and the y-axis those are the

vertical axis is the y-axis horizontal axis

is the x-axis now sometimes we’ll label these as t and s and

a whole bunch of different ways of labeling it but generically speaking

we’ll label this with an x and a y and we don’t need to label that

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right here is the origin and so what we’re going to have here is

different quadrants so this is going to be quadrant one

and it’s where x and y are both positive and we’re going to label this going

uh counterclockwise so this is quadrant two this is quadrant three

and this is quadrant four and so this will be where x is negative

so the x axis here the x’s are negative here and the y’s are positive

quadrant three the x’s are negative and the y’s are also negative

in quadrant four the x’s are positive and the y’s are negative

and so these are the quadrants right here and we divide the plane into four

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parts okay so you know we have the ordered pair and we have the signed distance

from the origin from the y-axis and the sine distance from the

so this is the signed distance you know from the x-axis

how far we got so the y is how far we’ve gone from the x-axis and it’s signed

it’s either going to be positive or negative okay so

and one last thing just to reiterate this is sometimes uh taken in context

because some this also looks like an open interval so you’ll know the

difference depending upon context if it’s a point or if it’s an interval

okay so very good so let’s plot some more points and so i want to just kind of

you know we plotted some points already but i just kind of want to

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um you know once you get past uh early stages and you’ve plotted some points

then you know because you’re gonna be plotting points throughout all of

pre-calculus all of calculus one two and three you know all throughout

pre-calculus and calculus and and continuing you’re gonna be plotting

points occasionally so the thing is is that when you first

start off you start drawing your axes and just like i did on the first example

i was meticulous i laid down take mark tick mark tick mark tick tick mark but

once you get past the initial stage of learning how to plot points

what’s going to be more important is what you’re doing with the points

so let me give you an example here suppose i want to graph y equals x squared

so this right here has a definitive shape to it

and if you don’t know that shape then you can plot points

and you can find and you can find enough you can find enough points to find the

shape however if you already know the shape

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and you want to plot some points on the curve or on the graph

then you don’t really want to lay down meticulously lay down tick marks so let

me let me give you an example so i know what this looks like it’s a quadratic

so i’ll just sketch the graph something like this roughly

got a little shaky there but so it looks something like that

and as you can see it’s a little rough down here but i think it’s okay i’ll put

in a point here point here so this will be the point here say one one

this would be the point here two what four and you see i’m not going to lay down

meticulously lay down and measure with a ruler because what’s more important is

that i actually get the shape of this that’s what’s crucial is to get the

shape now you can get the shape if you meticulously lay down tick marks but

it’s really unnecessary when you’re working out um complex problems you know

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this could be a cubic or quartet with several terms you want to get the shape

of that graph and you’re going to be learning pre-calculus and calculus to

help teach you methods to get the shape of things but eventually you also want

to plot points along uh those those curves of those graphs

so you know plotting points so you know i’ve seen students before they’ll

meticulously plot lots and lots of tick marks but it’s really unnecessary you

want to kind of just get the shape of things so for example what would this

point right here be so this would be four also and then this

would be minus two right here so that’ll be minus two so i labeled the

minus two and the four and so i plotted that point right there i don’t need this

right here i plotted this point right here and i’m not and you know

you know that’s so here’s a one right here

it says say this right right here is a half

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and then the height would be 1 4 there so one half squared would be 1 4. so the

idea is to make sure the shape of the graph

is important the the shape of the graph is more important than actually laying

down tick marks so that’s all i want to say here on this example here is that um

sometimes you’ll have to plot a point another example is sometimes you’ll have

to plot a point like five three thousand right so how would you plot that point

so you could just go here to five and then three thousand

and there’s the point right there so i’m not going to label

three thousand tick marks so i can get to that point right there right

so you know that’s all i’m saying is right there sometimes when you’re

plotting points the tick marks aren’t the most important

thing however when you’re first starting out yeah sure put the tick marks yeah

sure use graph paper and you know learn how to plot points the right way and

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then once you do that then you start understanding is as the map that you’re

studying becomes more complex that it’s the actual shape or the ideas behind the

mathematics and not the actual take marks along the graphs

okay so let’s go on now so let’s look at finding the coordinates of a point

and so i’m going to write the coordinates of the point

and the point is going to be given by some kind of description here

so the point is located three units to the left of the y-axis so let me just

draw this will be number one right here so three units to the left of the y-axis

so this is the y-axis so one two three and four units above the x-axis so four

units is up here and so this will be the point here minus

three and then four units above the x-axis so this will be the point here

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minus three so number one the point is minus three four so that’s the

right the coordinates of the point so it’s going to be three units to the left

of the y-axis and four units above the x-axis so

number one the point we’re looking at here right here is the coordinates is

minus three four now number two here the point is located eight units below

the x-axis so here we’re looking at eight units below the x-axis

so somewhere down here and then four units to the right of the y-axis

so four units to the right of so it’s gonna be four four units here and then

eight units down so here i’m going to be looking at the point here

four minus eight i think i’m going to label this one over here closer here

so there’s the point there this is about a four

and a minus three so number one the coordinates here and the number two

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located eight units below the x-axis so go down eight units and then four units

to the right of the y-axis okay now number three

so number three is the point is located five units below the x-axis so i’m

looking at the x-axis so i’m gonna go down five

and the coordinates of the point are equal so the coordinates of the point are

equal so if i’m going to go down five units below the x-axis

um that means i’m i’m going down to a negative five so now i want to have

something like a negative five negative so here’s the point here

so number three the point is going to be negative five negative five

so i got negative five because i’m going five units below the x-axis and the

coordinates of the point are equal so they’re both negative five right there

okay so good so number four here is the point is on the x-axis

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and 12 units to the left of the y-axis so way out here at 12 units so

i’ll just draw another one here scooting it over here

let’s do it about right here and then this is going to be -12

so the point is on the x-axis so that’s the x-axis right there and

it’s going to be 12 units to the left of the y-axis

so 12 units to the left of the axis so this point right here is better known as

minus twelve zero so that would be the coordinates of the

point in number four there okay very good let’s go on and now we’re

going to look at determining what quadrant the point lies in

so let’s look at some of these here so determine the quadrants in which x and y

is located so that the conditions are satisfied

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so number one here x is positive and y is negative so i’ll just draw some

coordinate axes here so for number one here the x is positive

so we’re going to be above the uh we’re going to be to the right of the y axis

and the y is negative so we’re going to be down here in quadrant four

so that’s number one right there is quadrant four number two

the x is negative four so let’s go over here to negative four

and the y is positive so we’re going to go up

so so number two we’re going to be in quadrant three

so this will be quadrant four right here and this will be quadrant three right

here okay so now number three x is greater than four

so let’s put a four right um yeah let’s put a four right about here

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and we’re looking greater than four so if the x is greater than four the y

could be either one positive or negative so this one right here i’m going to say

quadrant one or four so i’m going to say quadrant one or quadrant four

okay and the fourth one so the so negative x is greater than zero

meaning x is negative because we need a negative negative to be positive

so x is negative here that right there tells us that x is negative

so the x’s is over here and the y is also negative so we’re going to be

uh actually i said that’s quadrant three oh my bad that’s quadrant two

quadrant two right here for some weird reason i put that as quadrant three let

me go back and check this number two here so minus four and positive and

that’s definitely quadrant two if you’re confused by that good for you

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quadrant one quadrant two quadrant three so the answer for this right here so

negative x is is positive so x itself must be negative

for example negative 4 because it would give us a negative negative 4 which

would be positive right so x is negative and then the y is negative so so number

four here we’re going to be in quadrant three all right very good

and then the last one here is x times y is positive

so x and y are both positive or they’re both negative

so if they’re both positive we’re going to be in quadrant one

or if they’re both negative for example negative three times negative five

that would be a positive fifteen right so if they’re both negative then we

would be in quadrant three so quadrant three there

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so there’s number five is quadrant one or quadrant three

and this was quadrant one or quadrant two and that one quadrant two there

all right very good so if you have any questions or ideas

um for this video or other videos let me know in the comment section below

and don’t forget to check out the series functions and their graphs step-by-step

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