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in this episode you’ll learn the graphs of eight basic functions

these functions are some of the most commonly used and are called parent

functions let’s do some math [Music] everybody welcome back

all right so the first question is what are parent functions

and sometimes these are called toolbox functions um

whichever word you want to use these are some of the most commonly used

functions that you’ll see throughout precalculus and calculus and so it’s

very important to have these basic eight functions right here memorized in terms

of what the graphs look like and what are the domain and range i mean you just

want to have a very basic understanding of what these mean and then from these

we’ll build other more complicated graphs later on in the series oh that’s

right this uh episode is part of the series functions and their graphs

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step-by-step tutorials for beginners so i recommend checking out the complete

series the link is below in the description

and so we’re going to start off with the probably the most basic function of all

which is just a constant function so no matter what input you give the output is

the exact same it’s some constant some fixed constant

so a graph of something like this would look like this right here

so c could be positive or it could be negative or it could actually even be

zero so then you would just have a constant function right here so this

would be like y equals c for whatever c value is fixed so c is a constant

is a constant relative to the input right there so if

i input 2 i get out a constant i get out

of c if i input 3 i get out c if i input 0.9 i get out c

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if i input negative 1.3 i get out c no matter what the input is we get out c

so the domain of of this function right here is

all real numbers no matter what the input is

we get out c so you can input whatever you want

and the range and the range is just a single element so the range is

just the out the only output is c so the range is just the single set containing

the number c all right or another way you could write that is all real numbers

such that x is equal to c or in other words just

the set containing c the singleton set c all right and so there’s the first

function that you absolutely must know what it looks like it’s just a flat

horizontal line all right and so now the next one is the identity function

so let’s look at what the identity function looks like so it’s just y equals x

right and so that’s just the line going right through the origin the slope is

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one so y equals x so it has on the point for example one one you know 1.5 1.5

and so on or how about minus 3 and minus 3

right so whatever you put in is exactly what you get out

so this is the identity function and what’s the domain and range of the

identity function so the domain what can we put in

the only restriction is that you have to give out what you put in so there’s no

restriction at all so the domain is all real numbers and what’s the range

so the range is the output values well since i can input anything i want

i can output whatever whatever real number so the range is also all real

numbers so these are equal to each other because

you know you’re just getting in what you’re getting out you’re getting out

what you’re getting in all right so this is the identity

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function y equals x is the equation for it and here’s a sketch of a graph

all right once we go through these eight here uh

briefly then we’ll take a look at how to uh use python to how to just you know

how to plot them on python so it that’ll also be very short and very easy

all right so uh number three here is quadratic so you’ve probably seen

this before too y equals x squared so let’s just sketch the graph right here

y equals x squared and so you know let’s just make sure it’s symmetric here

all right that’s good enough so um this is going through 0 0

and here’s the point uh 2 4 right here 2 4 and we have here

let’s say about right here minus 2 and the height right there so that’ll be the

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point -2 4 right so this is symmetric with respect to the y-axis

and so yeah just a graph of y equals x squared here and so what is the domain

the domain is you know if you project the graph down onto the x-axis what part

is covered so that’s the domain the domain is all real numbers

or said differently what inputs can you square well you can square any real

number that you want there’s there’s no restriction upon what you can square you

can literally square any real number so the domain here is any real number all

real numbers what about the range so the range here is now if you project

the graph onto the y-axis so if we project this graph onto the y-axis this

is the part of the y-axis that will get covered and so that’s the range

the range is a set of output values now because the outputs always come from a

square right it’s always going to be positive so the range is

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we can say all real numbers such that x is greater than or equal to zero

or we could write it in interval notation starting from 0 and going to positive

infinity so if you want to write it as a set or interval notation

all right very good so number four a cubic function so let’s look at a

cubic function so y equals x to the third now maybe you don’t have this one

memorized but again you need to have all these memorized so that as soon as you

see this you can visualize that you can understand what it means in your in your

mind and this will be very advantageous for you for what’s coming up next and in

fact for all of precalculus and calculus so here we go here’s what a cubic looks

like it’s just going to come through here like this

and we talked about this function right here in the previous episode we

figured out that this is an odd function and that is in fact symmetric with

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respect to the x axis and so any point that we get on here x y

the point right here will be so that’ll be an x

and we’ll get a negative x right here and so this this will be the point right

here negative x negative y so this will be y right here

and this will be minus y right here so you know

for this equation right here we can plot a couple points for example we could

plot one eight sorry not one eight two eight and or we can plot one one

or two eight but you will have the corresponding point over here

which will be minus two minus eight and so that’ll that’ll be a point on the

graph also all right so this is the way the graph

is shaped nice and curvy i i like that shape right there

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and so what’s the domain so the domain is so again project this on to the x-axis

so project all these points down this keeps increasing so all these points

this is going to cover the whole x-axis you pick any point down here eventually

it’s going to climb up to it and it’s going to have a point on the graph so

the domain is going to be all real numbers this is just increasing the

whole way through the range is so what are the output values possible

so here we could not output negatives because we’re going to square it but

here if you know if you square if you cube a negative number you’re going to

get a negative number so you can get negative numbers out in fact you can get

any real number out and you can see that either just by looking at the rule here

or by in fact projecting the graph onto the y-axis

so if i project this point onto the y-axis and i project and i project and

what part of the y-axis is going to get covered it’s going to be the whole thing

the whole y-axis is covered so the range here is is all of our also all real

numbers all right very good so let’s look at our

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next one so now we’re going to look at the absolute value function

so the absolute value function isn’t so curvy

now actually the absolute value function i’m going to put it up here as a

piecewise function so this is very nice handy notation right here

but if you take a look at what the absolute value function is it’s

sometimes written as a piecewise function so you’re just going to output

an x or a minus x and you’re going to output x if it’s positive or 0

or you’re going to output a negative x if

if x is negative in other words if x is less than 0 then it’s negative and so

you need to do a negative of a negative and let’s make sure that this is just

the absolute value function right here so this is very handy nice notation but

in the end it’s just a piecewise function so for example what is f of 2

well 2 is greater than or equal to 0 so i’m going to output a 2. and that’s just

saying that the absolute values of 2 is 2. what if what if you input minus 2.

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now i check which ones control less than 0 so it’s going to be negative

negative 2 which is of course 2 which is of course absolute value of negative 2.

right so these these uh right here hopefully help you understand that

absolute value is really just a piecewise function so when it’s greater

than or equal to zero i just look like this line and when i’m less than zero i

just look like this line so you know the graph will look something like this

students often remember calling it a v or something like that

right so this would be the point zero zero

and for any point on here let’s call it x and let’s call this y

what will be the corresponding point over here it’ll be minus x and then the

same height y right here so this is symmetric with respect to the y axis

and so you know what’s the domain of this

of the absolute value function domain is so you can take the absolute value of

any real number you want and right so you can tell right here the

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domain is all real numbers if it’s greater than zero you input here if it’s

less than zero you input here so the domain is all real numbers

and so the range is now i’m looking at the output

so the output um you know that’s the whole point of the absolute value is to

make sure you get output of positive or zero so the range is all real numbers

such that x is greater than or equal to zero

so this is the same domain and range as uh number three here so number three

and number five have the same domain and range um but

the graphs are different this one’s curvy um this one you know just straight

lines right here this is a line right here with slope one this is a line right

here with slope minus one which you can tell from looking right here

all right and so there’s the absolute value function so you definitely want to

have this memorized in your mind um what it looks like and so what we’re

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going to be doing in the upcoming episodes we still got a couple more here

to go but just to give you a little preview

is um you know we’re going to take these

eight basic functions and we’re going to apply transformations to them and we’re

going to make more complicated functions for example

this is what the function right here y equals absolute value looks like but

what if you were to try to graph this right here so 2 absolute value of x

minus 3 and then plus 5. so having this graph

right here memorized what it looks like and what the domain and range is

now that we’re going to apply some transformations to it we’re going to

subtract the x by 3 we’re going to multiply by 2 and we’re going to add a

- so knowing what this looks like you should be able to mentally after some

practice be able to graph this in your mind um and and and i can do that really

quick because i know it’s just shifted right three and it’s shifted up five and

it’s been scaled by a factor of two so i can kind of visualize what this looks

like i can see it being shifted around in my mind and so that helps me make a

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sketch it helps me do a lot of work a lot quicker in any case we’re going to

get to all that in the upcoming episodes all right so here we go we got three

more to go here so number six we got the square root function so the square root

of x so you got to have this function right here memorized so here’s what it

looks like it’s just the square root of x

so this will be y equals square root of x and so it starts right here at 0 0

and the domain of is is going to be 0 to positive infinity so i’ll put it in

interval notation first i guess so we can write it as all real numbers such

that x is greater than or equal to zero you can take this you can take the

square root of zero it’s just zero and you can take the square root of any

positive real number and so what will the range be so the range here is

and so now we’re going to look along the y-axis what are the outputs well if

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you’re going to take the square root of it it’s it’s going to be positive right

so the range is zero to positive infinity

um or you could write it in set notation but it’s the same thing right so the

range and the domain here are the same um and so this has a nice arc to it

don’t draw this straight or you know it’s not shaped like a parabola

it’s not shaped like a line it has a nice curve to it now it doesn’t come

back down so don’t draw it like that it doesn’t do that it’s just going to keep

increasing it’s just that the further out you get

the slower it’s increasing but it’s always still increasing it’s a very

interesting very important type of function

all right and so now let’s look at the cube root function right here cube root

of x and so this will be our second to last

function right here so let’s look at a cube root of x here here we go

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all right so cube root of x all right cube root of x so this um

is different than i like to put this one right here next

to it so you can see the difference so this one looks like this

something like that it’s just increasing throughout

this right here looks different because so this one here is going smoothly

through here but this one right here is is coming in here and so instead of

doing this it’s going to come in here like this and it almost comes in like

it’s going in straight but it doesn’t go ever vertical so let me redo that it

doesn’t go vertical but it looks like it is going vertical

so it’s coming in there and it’s going like that

and so you know there’s very different kind of behavior here we can almost make

it even more more exaggerated here if we try to go in like really steep

it but it doesn’t ever go vertical so that’s important if it went vertical you

know like a vertical like vertical line then it would pass then it would fail

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the vertical line test right and this is the function and it just goes really

vertical like that the closer you get to zero [Music] right

and so it goes like that right there all

right so it’s increasing throughout just

like this one is but the behavior around the origin is very different so the

behavior of the origin around here is just it’s just going through there like

almost linear like a line when you’re really close to zero or something like

that but but this one right here goes very vertical right here very vertical

all right so you get better at sketching these actually when you get into

calculus because you go through uh the derivative and applications of

the derivative and you get skilled and you get a lot more tools in terms of how

to graph these functions right here so you actually learn these functions over

again but on a deeper level when you get to calculus so for the rest of

precalculus though you got to kind of you just got to kind of memorize them

but it’s important to plot some points and understand and you know maybe use

some technology to help you uh understand them but um here we can use domain is

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so the domain is all real numbers so if you project any point on the graph to

the x-axis you’re going to get the whole x-axis is covered so the domain is all

real numbers or said differently any real number that

you pick i can cube root it i can even cube root a negative number

right so we can cube root any any real number and what about the range

so now we’re going to project the graph onto the y axis

so if i take a point and project and take a point project if i take a point

project take a point project right so the whole y-axis is going to get covered

so the range is going to be all real numbers also

all right so there we go there’s the cube root of x all right so

um you know try to distinguish you know make your graphing skills good enough so

that you can distinguish between these two right here um okay so

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um you know for example on this graph right here what do we have we have 2 8

and on this graph we have 8 2 so this would be 8 2 so

you know when you get a small input the output is growing fast this is growing

much faster even just at two i’m already at eight right this is growing slower

but it is growing so all the way to eight i’m only at two

you know compare the difference between these two right here so you gotta have

them shaped differently you gotta you know show

between these two graphs especially if you have to graph them side by side

all right so now let’s go to um number eight here the last one

is going to be the reciprocal function right here so 1 over x

so let’s sketch this real quick 1 over x

now this one is very interesting to me i love this function right here

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and it has a vertical isotope a zero and it has a horizontal isotope of

the x-axis y equals zero and i dash those in because they help

guide me in terms of shaping my graph so in other words how can you communicate

something to someone when you say that it’s decreasing but it never crosses the

x-axis so i draw this dashed line and this is going to look like this and

it’s going to look like this right here so let’s look at the domain of this

right here the domain is all real numbers in other words what can

we input we can input any number we want except zero so the domain is

all real numbers except x is not zero or said differently we could write in

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interval notation minus infinity to zero union zero to positive infinity

and make sure these are rounded because we’re not going to include zero

and so we can see that by projecting to the x-axis if i take this point and

project it to the x-axis and this point in this point and if you look at all the

points on the graph the whole x-axis is going to get covered when i project so

i’m projecting to the x-axis and i get covered project to the x you know just

keep projecting to the x-axis what part of the x-axis is going to get covered

all of it except for zero there’s not going to be any point projecting onto the

x-axis at zero so that’s why that’s a graphical way of

thinking about the domain and what about the range so let’s put the range right

here the range is so now i’m projecting on to the y-axis for the range so the

range is the output values so now when i pick a point on the graph i project to

the y-axis pick another point pick another point and just keep projecting

to the y-axis and ask yourself what part

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of the y-axis is going to get covered so it’s going to be the whole y-axis

all real numbers and now i use an x there but maybe you want

to use a y all real numbers y such that y well there’s nothing that’s

going to be projecting all the way to zero so i’m going to say y is not zero

here or you know we could write it like this in here in interval notation also

so the domain and range here are actually the same it’s just that one occurs uh

by projecting to the y-axis and one occurs by projection to the x-axis if

you want to think in terms of geometry right there

so anyways let’s get a good sketch back and i just want to make sure that

we’re okay on these right here i think this is really important to know

this graph right here when you get to calculus you’ll see this graph is an

example of several things so let’s just double check that the domain and range

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are these two things here without thinking about projections so i think

it’s pretty clear that this is the domain because if you try to input zero

i think probably most students have heard hey you cannot divide by zero but

what about the range how can you just look at this right here and know the

range is all real numbers except zero so let’s think about this right here like

zero over ten is zero zero over minus one hundred is zero

zero over anything i put down here except except i can’t put zero but if

any other number any other real number i put down here i’m gonna get out zero

because i already have zero on the top so the only way to get zero is if you

have zero so we don’t have zero up here so there’s

no way to output zero from all of this no matter what you input for x there’s

no way to output zero all right so there’s the domain and

range of the one over x function right there

and now what i want to do is just to briefly take us through how to graph

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these uh using a computer and we’re going to use

that in in an upcoming episode also when

we start doing transformations but right

now i just want to get a quick sketch of the graph we’ve already done them over

here by hand but i just want you to see how to do them also using python so

we’re going to open up a python notebook and if you don’t know how to do that

there’s a link below in the description which will take you to a free python

notebook that you can open up and yeah so let’s get started on that so

here we go let me zoom in here a little bit and i’m

actually going to move up here so let’s look at our setup here that we got

now you don’t need this cell right here called setup and what’s important are

the inputs you need to input whatever is in here so

we’re going to input and then now for this setup right here for this for what

we’re about to do i need to import some packages so these packages make it very

easy to use python so this is the python programming language and i’m going to

import this package right here and i’m going to import another package right

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here and another one numpy and math and so to uh once you type those up exactly

as they are then you just hit shift enter and you execute that cell

now in my precalculus series here uh i like to customize my axes when i sketch

graphs so this is going to be a function which is going to take in some axes and

it’s going to graph the axes the way or plot the axes

the way i like them so let’s just execute that cell so type that up if you

watch previous episodes you probably already have that done

and then i’m going to use this function right here called plot function and we

also use this function right here in a previous episode so it just takes in a

function and it can take in a minimum value but the

defaults -10 it can take a maximum value

but the default is 10 and it takes in an increment right here which you can take

pass in if you don’t like that default value so these are default values here

and i’m going to plot it i’m going to give a domain i’m going to give some

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outputs and i’m going to plot it and then i’m going to show that plot

all right so there’s a brief description of that function right there

all right so now let’s look at the parent functions let’s look at each one

of these eight individually now actually i didn’t even bother to do

the constant function so i just started with the identity function right here

and so the identity function takes in an x and it returns an x and that’s just a

very basic function so i’m going to plot the identity

function right here so plot function and i’m going to give it the function

and so there it gives me the identity function right there now for each one of

these parent functions that we’re going to go through

we want to pay attention to the window so for this identity function minus 10

to 10 that shows the shape of it just as good

as any other window so i’m just going to leave that the way it is

so now let’s look at the quadratic function y equals x squared

so i’m going to enter that definition in right there

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that’s going to define what quadratic function is

you input an x i output x squared so now we’re going to plot this function right

here and we’re going to look for a good window and again from -10 to 10 the

default window yeah it looks great there’s the nice curvy shape of it

and i like that output right there so now let’s look at the cubic function

so this will be y equals x to the third i input x i output x to the third

so let’s execute this function right here cubic function

now i’m going to plot it so i hit shift enter

so plot function cubic function and we get this nice shape in here

now perhaps it’s not very clear what’s happening around here so i think we can

change this window right here maybe we’ll go to minus one to one

and if it looks a little choppy then we can actually also increase the

increments right there i don’t know if you can tell if it looks choppy or not

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so i’m going to add an extra zero into the fourth argument function x-men

x-max and then increment so i feel like it looks a little bit choppy right here

and if i execute that now it looks a lot smoother in there so the increments uh

kind of determine how smooth the graph is there so i think that’s a good graph

there alright so now let’s go on to the absolute value function right here

so i’m going to define the absolute value of x function

and to be honest there’s already an absolute value function built in so i’m

just going to use that so this function doesn’t really do very much

but i’m going to plot it and i’m going to look for a good window

so i’m going to use the default minus 10 to 10

and there we go and so this looks just like any other as good as any other

window minus 10 to 10. so there we go there’s a nice uh sketch

of it uh it’s symmetric so you know any symmetric window would work there

alright so square root function again there’s a built-in square root

00:28

function into numpy and that’s one of the packages we

added at the very beginning in the setup so i’m going to define the square root

function input in x output the square root of x

so shift enter and now let’s plot it now i do want to change my window i don’t

want to go from -10 to 10 because well you’re not going to be taking the neg

the square root of any negative number so i want to start from zero and i’ll

just go to 10 and so there we get a nice shape of the graph right there

so this is the square root of x so notice how x squared goes up faster

than square root of x and so we’ll see that play out when we

do the cube root function also so here’s the cubic function it’s going up pretty

fast right here now and the further out you go the faster it goes but

now let’s look at the cube root function

here so the cube root function i’m going to input an x

and again there’s already a built-in cube root function so square root

00:29

function was sqrt and this is cube root function so let’s uh shift enter that

and now let’s plot the cube root function right here and now we’ll get

this nice shaping right here and you can tell right in here it’s getting pretty

vertical but it never gets vertical right because it’s a function and it

passes the vertical line test so it’s just the closer you get to zero the

faster it’s increasing that’s a pretty interesting function i love this

function then the last one is the reciprocal function

and there is a built-in reciprocal function so i just use that numpy

package and i use the reciprocal there and so that function has been executed

and so now we want to go plot it now i’m actually going to plot this on

and make two plots of it as you can see right down here so the first one i’m

going to plot positive so 0.1 to 3 so i plot 0.1 to 3

and i make it so it’s not very choppy at all so i increase my increments right

00:30

there and i’m also going to plot at the same time the same function

but now on the second plot i’m going to plot from -3 to 0.1

so that’ll be right about there and then i’m going to use the same kind of

increment right there so let’s execute these two

function evaluations these two plot functions and then we get out these two

plots right here so now you can see on the right side of the origin you get

this and then on the left side of the origin you get here right here

so this has some isotopes here you know the the main graph is consists

of both of them together we have an isotope we have an isotope

and on the right side we get this which is this part right here and then

on the left side we get this part right here which is this right here

so make sure that you don’t make it look like it went straight down because that

would be incorrect as it does not go vertical

00:31

also make make sure that you don’t make it look like it curves back down

sometimes i see students do that it doesn’t do that

it’s not increasing here at all it’s always decreasing on this interval

same thing here make sure and don’t do that

it’s it’s never increasing at all it’s just always decreasing

all right so there we go there is the um parent functions right there all eight

of them now in the upcoming video uh the upcoming episodes i will refresh your

memory on what those parent functions are and we’ll start doing some

transformations so you’ll see how useful it is to have these eight graphs

memorized so that we’ll be able to plot so many more functions just by thinking

about it and we’ll do it by hand for sure but you’ll be able to actually

after a while after you get enough practice in be able to actually

visualize how a lot of graphs look like just from knowing these basic functions

00:32

well if you enjoyed the video today and i enjoyed making it i like to hear your

comments below and until the next episode well i’ll see you then

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