Parent Functions (and Their Transformations)

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

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

00:00
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

00:01
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

00:02
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

00:03
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

00:04
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

00:05
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

00:06
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

00:07
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

00:08
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

00:09
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.

00:10
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

00:11
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

00:12
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

  1. 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

00:13
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

00:14
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

00:15
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

00:16
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

00:17
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

00:18
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

00:19
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

00:20
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

00:21
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

00:22
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

00:23
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

00:24
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

00:25
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

00:26
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

00:27
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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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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