Testing for Symmetry (the Graph of an Equation)

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

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

00:00
this episode you’ll learn how to test for symmetry with respect to the x-axis
the y-axis and the origin [Music] a graph is symmetrical with respect to
the x-axis let’s talk about the x-axis first so if whenever x a point x y is on
the graph then the point x minus y is also on the graph
so the typical example i like to think about in my mind is something that looks
like a sideways parabola or y squared equals x
so if i pick a arbitrary point on this graph right here call it x y
then the x minus y would also be on the graph so this would be the point here
say x y and this would be the point down here x minus y
so if i can pick any point on here and then the point x minus y is also on the

00:01
graph or if i pick any point down here then the negative of the negative will
be positive right so this graph is symmetric with respect to the x axis
and so now we have uh symmetry with respect to the y-axis
so then i like to think about this as my example here um just you know the
y equals the x squared we’ll draw something that looks like that
and so now if i pick a point on here x y any any arbitrary point
then the point also is on here minus x minus xy right so
let’s just draw a little bigger here minus xy would also be on the graph
so um this would be symmetric with respect to the y axis

00:02
and then the last one we’re going to consider is symmetry with respect to the
origin if x y is on the graph then minus x minus y is also on the graph
and the typical one there we can think about here is just the line right
through the origin y equals x so if i pick a point on here call it x y
for example one one then the point here minus one minus one or
any random point x y so minus x minus y would also be on the graph so this would
be symmetric with respect to the origin and so i you know when i’m reading these
definitions here i like to have these illustrations in my mind just to keep
track of and make sure that i’m understanding things
um that’s good practice for anything uh you know have examples in your mind

00:03
when you’re looking at definitions so x-axis y-axis and origin
and so now let’s look at an example so let’s test for symmetry here
when i look at this equation here um f of x equals x when i look at this
function here right so that’s the one we just looked
at um y equals x and that was uh symmetric with respect to the origin only
so number one right here we would say you know there’s the graph and it’s
symmetric with respect to the origin only symmetric with respect to origin
number two y equals x squared and so we also looked at that one a minute ago
y equals x squared that’s symmetric with respect to the y axis

00:04
i’ll just say symmetric with respect to y axis and then so for number three
what about x to the third so maybe you know what x to the third looks like
so i wanted to do a couple of examples of symmetry testing for symmetry when you
know what the functions look like we know what these functions look like already
y equals x to the third and just by looking at the function
it’s symmetric with respect to the origin so symmetric with respect to origin
and the fourth one is the absolute value of x and so
this will be symmetric with respect to the the y axis okay so
in case you can’t see that there number four symmetric with respect to

00:05
the y axis um and then just you know what would be
symmetric could you think of what would be symmetric with respect to all three
um i know it’s not one of the ones down here but just you know what what would
the circle be symmetric to right so that would be
symmetric for example the unit circle that would be symmetric with respect to
all three it would be symmetric with respect to
the y axis because anything over here can be folded over and and graphed over
here or the x-axis anything up here anything up here can be folded down or
you can fold this part up or symmetric with respect to the origin and we’ll
look at the circle as an example here in a second
but let’s look at some more common functions that we’re used to
to knowing the shape of so let’s look at the square root of x here now
what is the square root of x look like so
number one here will be square root of x and that looks something like this

00:06
and so that’s not symmetrical with respect to the y to the y axis if it was
there would be some other part over here it’s not symmetric with respect to the
x-axis and it’s not so if it if it had something that down here right so this
is not symmetric so this is this is none right so not
symmetric with respect to any of them number two how about the cube root of x
so the cube root of x looks something like this
and it is symmetric with respect to the origin so we’ll just say symmetric
with respect to origin uh number three what we’re going to look at for number
three here how about one over x so that looks something like this
there’s a vertical isotope here and there’s a horizontal isotope here

00:07
it’s not symmetric with respect to the x-axis
there’s not none not another part down here it’s not symmetric with respect to
the y-axis these are isotopes so i’m just drawing a rough sketch real quick but
so it is symmetric with respect to the origin symmetric with respect to
the origin and number four we look at number four here one over x squared
so that looks something like this so it also has a vertical isotope at zero
but now this shape looks like this and this these uh negative x’s um
give negative y’s so it’s down here but when it’s squared
then the y’s become positive it looks like that

00:08
and this is the y-axis right so this is symmetric with respect to the y-axis
so good there’s four more examples there and these are just basic uh graphs that
you probably already know how they’re shaped and just to run through them and
just to see them because usually when someone asks you to test for
symmetry you really want to test for it algebraically
knowing the shape of them can in some cases just be enough for your own
knowledge but if someone’s actually going to ask you
hey show this is symmetric then you want to algebraically test for symmetry so
let’s see that so let’s test for symmetry algebraically so this is um
an equation here maybe you don’t know how the graph looks so testing for
symmetry can really cut down on the amount of work it takes to make a good

00:09
sketch of it so we’re going to test for symmetry so test for symmetry and
so let’s test for the x-axis let’s test for the x-axis here so
i’m going to replace y with minus y so our equation is x y equals x to the
fourth minus uh x squared plus three and so let’s replace y with a minus y
and this is what we get we have to ask are these the same
and the answer is no they’re not the same they’re clearly different so let’s
say not symmetric and so this is how you could test for
symmetry with respect to the x-axis now let’s test for symmetry test for y-axis
so now i’m going to replace x with negative x and so i’m going to start with my

00:10
original equation or function and wherever i see an x i’m going to put
a negative x so this x here is going to go to negative x
and this x is going to go to a negative x make sure and put parentheses in there
because wherever the x is you want to replace that whole x with a minus x and
now i’m going to simplify is there anything to simplify here over here
there really wasn’t anything to simplify
but minus x to the fourth is just better known as x to the fourth
and this is minus x squared that just means minus x times minus x right so
that’s positive x we still have a minus here so in fact we get the same equation
right here we this is our original we replaced x with negative x everywhere
and we simplified we got back to the original so we’re going to say yes
symmetry so it’s symmetric with respect to the y axis now we can test for

00:11
the origin and so to do that we’re going to replace x with negative x and y with
negative y and so i’m going to start with my original here’s what it is
all right so i’m going to have negative y and then i’m going to have negative x
to the fourth and then minus negative x squared and then plus three
and that’s going to be negative y and that’s going to be x to the fourth minus
x squared plus three and we got to compare our starting with
our ending are they the same here’s the starting one that’s just the original
and then here’s what we obtained after after doing x with negative x and y with
negative y they’re not the same so i’m going to say not symmetric
with respect to the origin and so now when we’re sketching this
graph right here knowing that it’s symmetric with respect to the y axis
that can uh help cut down the time of sketching this graph because

00:12
whatever this graph looks like it’s going to be symmetric with respect
to the y-axis so i just need to plot points over here on this side of the
y-axis for example and then whatever points over here i can match it over here
okay so there’s testing symmetric testing for cement symmetry algebraically
let’s do another one though and so let’s go with this one right here
so this this time we don’t have a function we have an equation
and so i’m going to do test testing for x-axis and i’m going to do testing for
y-axis and i’m going to do testing for the origin all right oops

00:13
i wasn’t paying attention there so here i go i’m just going to say x-axis y-axis
and origin here all right so when i’m testing with
respect to the x axis i need to replace y with negative y
so i’m going to go here with negative y squared here plus 10 equals zero
and so i replace the y with negative y now i’m going to do that squared that’s
x y squared plus 10 equals zero and so i got the same so yes yes symmetric
now about what about the y axis so now for the y-axis i’m going to
replace an x with a negative x so here i’m going to get a negative x
times y squared plus 10 equals zero and there’s nothing to do here there’s
nothing there’s there’s nothing to simplify so i’m going to say no not symmetric

00:14
and what about the origin so now everywhere so negative x
and then also the negative y and here this will be negative y squared plus 10
[Music] and then so not symmetric here either
okay so there’s another example is this one is symmetric with respect to the
x-axis so i would just need to us plot points if i was sketching this graph
here above the x-axis and then i’d be able to find all the points below
and so that could help you understand the shape of this graph better
let’s do another one here we go so this one is x equals
x equals y squared minus five and so let’s test uh with respect to the
x axis here so we’re going to place y with negative y

00:15
and so we’re going to get x equals minus y square plus 5 minus 5.
all right and we’re going to do the squared here so that’s x equals y
squared minus 5. and so yes this is symmetric with respect to the x-axis
symmetric here we go now for the y-axis here the y-axis i’m going to replace
an x with a negative x so i’m gonna get negative x equals y squared minus five
and they’re not the same so no so not symmetric with respect to the y axis
and then how about let’s test the origin here so i’m going to say negative x
equals negative y squared minus five so i get negative x equals y squared
minus five and so i’m going to say not symmetric

00:16
not symmetric with respect to the origin okay and so then we’re going to
do one last example here test for symmetry and this is our unit circle
here so let’s try the unit circle is it symmetric with respect to the
x-axis y-axis in origin so we sketch this before just drawing a circle and we
noticed it was symmetric with respect to all three let’s see how the tests work
out so test x axis so i’m going to replace the y with a negative y
so that should be negative y squared equals one so x squared plus y squared
equals one so that works so test the y axis symmetry
and so now we’ll use a negative x and so that will give us the x squared

00:17
plus y squared equals one so it is symmetric with respect to the y axis
and now i’ll do both so test origin so this is symmetry with respect to the
origin so i’m going to use a negative x everywhere i see an x
and a negative y everywhere i see a y and i simplify and
we get the same exact equation after simplifying we get the same equation so
yes it is it is symmetric with respect to the origin
and so that’s it for today’s video so if you like the video go ahead and
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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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