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How to Tell if a Function is Even or Odd From Its Graph

hey guys I just want to do a quick video

today about how to determine if a

function is odd even or neither by

looking at its graph so you may not have

the function itself to work with but now

we're given a graph and there's three

rules for this there's the y axis which

is vertical symmetry if if that is if

there's y axis symmetry the function is

even which means you can reflect over

the y-axis and the graph would be the

same and there's origin symmetry which

is rotational symmetry if the function

is odd which means we could turn the

graph like 180 degrees and it might be

the exact same graph you had before so

just think about it you could turn the

graph on its side and it would still

equal the same thing and then if there's

neither y-axis symmetry or origin

symmetry the function is neither even

nor odd so now the best way to do this

is just to look through examples so

we've got this parabola here facing down

and let's just go through the symmetries

does it have y-axis symmetry well if I

reflect it over the y-axis so if I was

to turn my page in half at this y-axis

right here this one right there at this

line would the graph look the exact same

and that answer is yes

because I could rotate it right here and

have the same graph it means that it is

y-axis symmetry which means the function

is even alright so now we got the next

one um does this have y-axis symmetry

could I reflect over the y-axis did have

the same function no if I was to reflect

over the y-axis this axis right here I

would be left with this right here and I

would have nothing over there

so there's definitely not y-axis

symmetry all right now let's think about

origin rotational symmetry so what the

origin symmetry is means I could just

turn it on its side at the origin and it

would equal itself at some point so

let's think about it well if I turned it

90 degrees I'd be left with something

like this if I turned it 180 I'd be left

just something like this if I turn it

again I'm left with something like this

and let me erase these and then finally

if I turned it 360 degrees of course it

would be the same so that doesn't even

count so there's no origin symmetry I

can't rotate this graph

and get and be left with the exact same

one and now I'll show you an example of

what origin symmetry does look like is

this right here we can turn it 90

degrees and what would we be left with

well I'd be looking like let me think

it'd be like this I think and then like

this or something so that wouldn't be

yet but then if we turn it 180 degrees

we'd be left so we're turning it right

over here

we'd be left with the exact same graph

that we're looking at right now and that

means that there's origin symmetry so it

is odd so this function is odd this

function is even and because this

function had neither y-axis symmetry or

origin symmetry this function is neither

so those are the three rules if you can

memorize this you'd be fine on any test

or anything that you're gonna have to do

just be careful with the origin symmetry

it's sometimes hard to imagine you

physically rotating the graph but this

as you knows your y equals x cubed so

just know that a cubic function is going

to be odd and know that the square root

function is neither one just in case and

this is also a parabola so parabolas are

generally going to be even so the best

way to do it is just to see practice

problems and hope this was able to help

let me know if you have any questions