How To Plot Points in the Cartesian Plane (the RIGHT Way)

Video Series: Functions and Their Graphs (Step-by-Step Tutorials for Precalculus)

(D4M) — Here is the video transcript for this video.

00:00
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

00:01
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

00:02
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

00:04
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

00:06
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

00:08
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

00:09
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

00:10
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

00:11
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

00:12
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

00:13
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

00:14
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

00:15
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

00:16
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

00:17
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

00:18
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

00:19
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

00:20
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

00:21
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
tutorials for pre-calculus and i want to say thank you for watching and have a
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About The Author
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David A. Smith (Dave)

Mathematics Educator

David A. Smith is the CEO and founder of Dave4Math. His background is in mathematics (B.S. & M.S. in Mathematics), computer science, and undergraduate teaching (15+ years). With extensive experience in higher education and a passion for learning, his professional and academic careers revolve around advancing knowledge for himself and others. His work helps others learn about subjects that can help them in their personal and professional lives.

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