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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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