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hi everyone welcome back i’m dave this video is uh how to plot points in

the cartesian plane the easy way in this episode you’ll learn how to plot points

in the cartesian plane the easy way by using a computer

um first i want to mention before we get

started that this is this videos part of the series functions in their graphs

step-by-step tutorials for pre-calculus so the link is for the full series is

below in the description so um in this episode we’re going to talk about

setting up axes for a plot and we’re going to talk about how to plot points

and we’re going to talk about graphing functions now we’ve already talked about

these things right here in previous videos but in this video we’re going to

do things on the computer in particular you can follow along by opening up your

own python notebook link is below in the description and it

only takes a moment to get started but you’ll see how powerful and how how

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useful it is and so right now let’s go ahead and get started [Music]

alright so at first was we’re going to need some axes so typically when you’re

in a pre calculus course and you’re working in the cartesian coordinate system

we’re going to draw some axes a particular way so in in a in a

pre-calculus course you would see something like this

where you would have the x-axis and the y-axis and you usually have arrows over

here to indicate the positive direction and the positive direction up here

so we need to have something like this uh whenever we sketch our graphs

because computer graphics software often don’t have axes that look like that so

i’m going to show you how to make those axes uh using a computer

and so let’s go ahead and see how to do that

so we’re going to look at a python uh notebook right here and first thing we

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want to do is set up our python notebook by importing some

packages here so to execute the cell i’m gonna hit shift enter

and again you can use the link just below in the description to follow along

in your own python notebook and so right here um we’re going to have

a figure this is what a default figure looks like and and in a python notebook

so uh first thing is i’m going to declare what a figure is i’m going to

declare a figure and then i’m going to declare some axes using this plt using

the matplotlib package and i’m going to declare um

some axes and then i’m going to show it right and so if we execute the cell we

get this cell right here and this is what it looks like by default so as you

can see we have a border around here and we have some tick marks right here so

um i want to change it and make this look like something that you would see

in a traditional pre-calculus course so um we’re gonna customize this what

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we’re gonna do is we’re still gonna declare figure

we’re still gonna declare a set of axes and then what we’re going to do is we’re

going to customize the way the the the plot looks so we’re going to say the

left and the bottom this is the left and this is the bottom we want that to go

through zero zero still but i want to turn off the right one and

the left one sorry the top one and the right one

and what i’m also going to do is i’m going to put some background color on

here this one up here’s why the background is why

if you don’t want to do that if you don’t want to customize your background

color then you can just leave off those two lines right there but i’m going to

go ahead and do that and then i want to put some arrows on

here as you can see what it’s going to look like down here after we execute

this cell right here so these two lines right here

put arrows on here and by the way we here’s the way we write comments in

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python right there so i’m going to execute this cell right

here and as you can see right here we get something that looks much more

traditional than this right up here so this is the way i’m going to

plot our functions and our points and precalculus in these in this series here

so in order to make this quicker though i’m going to go ahead and define a

function so def we’re defined and the name of this function is uh pc for

precalculus pc underscore axes and then the input for this function

is going to be whatever axes i’m working

on and then here’s the customization and this is the same customization that we

just saw up right here so i’m going to declare which ones to

show which ones not to show and where them to go through i’m going to use the

background color and i’m going to have some arrows so once we define this

function here by hitting shift enter now it becomes a lot quicker and easier

to plot some axes right here so again on every

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time i want to make a plot i just want to declare figure

i want to declare some axes and then i want to customize the axes and then i

want to show show my figure so let’s execute that and as you can see now we

get these standard uh looking axes that you would see in a pre-calculus class

and so now let’s go and look at plotting some points here [Music]

okay so as you can see over here we have this example over here and it says plot

these points and so i have a whole bunch of points here here’s an x and a y

here’s an x and a y and in the last video i talked about how to do this

um by hand and when we did this by hand you know we

had to to sketch the axes out and make the tick marks and so on

i guess we can go ahead and do one right here

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so we can graph this one right here is the easiest one zero zeros right there’s

the origin one tick mark one tick mark here’s the point one one

we have the x and we have the y and so on so we can go through this list

right here and plot all these points here

here’s how we would do this by computer so now we have uh we’re going to declare

figure we’re going to declare a set of axes we’re going to customize the axes

and at the end we’re going to show it so those three are those three lines are

the same what’s new well in this example here i’m going to

say these are the x’s right here so instead of drawing them horizontally now

we’re going to type them up horizontally into an array we’re going to separate

each one by comma and we’re going to do the y’s

and so you know you can think of this as a minus three nine a minus two four

minus one one so there’s the points right there and then we’re gonna use

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something called a scatter plot so scatter plot will not connect the

points at all they’re just points all by themselves so

let’s go ahead and execute that cell and then we get the points right here

so this is can plot them basically as fast as you can type in the points right

here so this is a really nice fast way of plotting points here

and let’s go ahead and see another example so in this example here

you know we have some x’s we have some y’s and so let’s go ahead and plot these

points here so when i plot these points here i’m going to you know use the same

i’m going to declare figure set of axes i’m going to customize my axes we’re

going to type in all the points and then we’re going to type in scatter dot x y

scatter x y on the axis that we’re declaring

and then we can show it and then we get all the points here so let’s execute

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that cell there there we go and there’s all the points

right there there’s the scatter plot again scatter plot just means we’re not

going to be connecting them in in with line segments

and that’s in contrast to what we’re going to do next which is graphing

functions so there’s the x’s and there’s the y’s and there’s the scatter plot

all right great so now let’s look at graphing functions so let’s see that

okay so when we’re graphing functions we’re going to do something very similar

we’re going to declare a figure we’re going to declare a set of axes and at

the end we’re going to show the plot so how and we’re going to customize the

plot right so how does it change so here you can see this has changed

dramatically here because i no longer am typing in a list of points um so my x’s

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are made automatically and my y’s are made automatically according to the rule

so notice here for exponent 2 we’re using this notation here

and what about the x’s so the x’s are gonna start from minus five to five so

right here we have minus five all the way to five and i’m gonna work in steps

in point zero one so these are very small steps right here

and so let’s go ahead and execute this cell right here

and then we can see we have a nice smooth looking curve right there

now i want to experiment with this here and and show you how this changes so

let’s go from point zero one where it looks nice and smooth let’s go to a one

and now you can see it’s a little bit more choppier you can kind of see these

points right here where it gets really choppy

and we can make it even more obvious by say moving that to a two [Music]

see what it’s doing is it’s not only plotting all of these points but it’s

connecting them with a line segment so that’s how plot difference is

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different than the scatter that we used before

so this is going to go from -5 to -3 because it’s in steps of two it’s going

to plot this point it’s going to plot that point and it’s going to connect it

with a line segment so now it doesn’t look as smooth

so to make it look smooth we can plot lots lots of points

there we go now it’s very nice and smooth

and so this is how we plot this function right here y equals x squared

and so that looks really nice and smooth and we have our axes

we could label our axes if we want [Music] so there we go

so let’s look at another example so you may have known what this previous

function looks like y equals x squared right so you may know what that looks

like what about this one do you know what this one looks like

so this one is you know perhaps more difficult to graph by hand

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but by computer it’s actually pretty much the same degree difficulty

we declare figure we declare a set of axes we’re going to customize the axes

we’re going to show the plot so here we need to um we need to have a [Music]

starting place here so i’m going to say -3

let’s go with -3 and then all the way to 2 and i’m going to go in small little

steps 0.01 and then here’s how i type up the function right we have an exponent

for the four we have a minus three x squared so i

have a minus three i like to i like to use spaces

but not with exponents it’s just my way of writing it

um let’s execute this and so now we get this nice plot right here

however this may not be the best um plot for you for me it stops here at

60 right so let’s let’s say let’s experiment with this a

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little bit that’s how i came up with these numbers here minus two to two

i think this looks the best right here so minus two to two

i’m going to use that as my window to sketch this graph right here we have a

nice clean shape right here looks very symmetric

okay so that does it for this video so i hope that you like this video and i

hope that you go ahead and subscribe and if you have any questions let me

know in the section below and i look forward to seeing you next time

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