- WELCOME TO A LESSON ON SYMMETRY ABOUT THE X AXIS,
Y AXIS, AND THE ORIGIN.
THE GOALS OF THE VIDEO ARE TO DEFINE
THE DIFFERENT TYPES OF SYMMETRY
AND THEN ALSO TO DETERMINE IF THE GRAPH OF THE GIVEN EQUATION
WOULD BE SYMMETRICAL ABOUT THE X AXIS, Y AXIS, AND THE ORIGIN.
LET'S START BY DISCUSSING SYMMETRY ABOUT THE X AXIS.
THE GRAPH OF AN EQUATION
IS SYMMETRICAL ABOUT THE X AXIS
IF WHENEVER X, Y IS ON THE GRAPH, SO IS X, -Y.
SO FOR EXAMPLE, LOOKING AT THIS GRAPH,
IF WE SELECTED THIS POINT HERE AND LABELED IT X, Y
NOTICE HOW THIS GRAPH WOULD ALSO CONTAIN THE POINT DOWN HERE,
WHICH WOULD HAVE COORDINATES X, -Y.
AND THIS WOULD BE TRUE FOR ANY TWO POINTS ON THIS GRAPH
BECAUSE IT HAS X AXIS SYMMETRY.
THERE IS ALSO A WAY TO VERIFY X AXIS SYMMETRY VISUALLY.
IF WE FOLD THIS GRAPH ACROSS THE X AXIS
OR ACROSS THIS HORIZONTAL LINE,
NOTICE HOW THE TOP HALF AND THE BOTTOM HALF
WOULD MATCH UP PERFECTLY.
IF WE FOLD THIS GRAPH DOWN ACROSS THE X AXIS,
NOTICE HOW THE TOP WOULD MATCH UP PERFECTLY WITH THE BOTTOM.
AND THERE'S ALSO A WAY TO VERIFY X AXIS SYMMETRY ALGEBRAICALLY.
THE GRAPH OF AN EQUATION
WILL HAVE SYMMETRY ABOUT THE X AXIS
IF WE GET AN EQUIVALENT EQUATION WHEN Y IS REPLACED WITH -Y.
SO FOR EXAMPLE, THIS GRAPH HAS AN EQUATION X = Y SQUARED.
IF WE REPLACE Y WITH -Y
WE'D HAVE THE EQUATION X = -Y SQUARED
AND SINCE A NEGATIVE TIMES A NEGATIVE IS POSITIVE,
WHILE RAISING A NEGATIVE TO AN EVEN POWER IS ALWAYS POSITIVE,
THIS WOULD SIMPLIFY TO X = Y SQUARED,
WHICH IS EQUIVALENT TO THE ORIGINAL EQUATION,
WHICH VERIFIES THE GRAPH WOULD HAVE X AXIS SYMMETRY.
NOW LET'S TALK ABOUT Y AXIS SYMMETRY.
THE GRAPH OF AN EQUATION IS SYMMETRICAL ABOUT THE Y AXIS
IF WHENEVER THE POINT X, Y IS ON THE GRAPH,
THEN SO IS THE POINT -X, Y.
SO AGAIN, FOR EXAMPLE,
IF WE LABEL THIS POINT HERE WITH COORDINATES X, Y,
NOTICE HOW THIS GRAPH WOULD ALSO
CONTAIN THE POINT ON THE LEFT WITH COORDINATES -X, Y.
AND AGAIN,
THIS WOULD BE TRUE FOR ANY TWO POINTS ON THIS GRAPH
BECAUSE IT DOES HAVE Y AXIS SYMMETRY.
VISUALLY, IF WE FOLD THIS GRAPH ACROSS THE Y AXIS
OR THIS VERTICAL LINE HERE,
ONE HALF OF THE GRAPH
WOULD MATCH UP PERFECTLY WITH THE OTHER HALF.
FOR EXAMPLE,
IF YOU FOLDED THIS TO THE LEFT IN THIS DIRECTION,
NOTICE HOW THE RIGHT SIDE
WOULD MATCH UP PERFECTLY WITH THE LEFT SIDE.
AGAIN, VISUALLY VERIFYING
WE HAVE Y AXIS SYMMETRY AND THEN ALGEBRAICALLY,
THE GRAPH OF AN EQUATION WILL HAVE SYMMETRY ABOUT THE Y AXIS
IF WE GET AN EQUIVALENT EQUATION WHEN REPLACING X WITH -X.
SO THE EQUATION OF THIS GRAPH IS Y = X SQUARED.
IF WE REPLACE X WITH -X
WE WOULD HAVE THE EQUATION Y = -X SQUARED,
BUT AGAIN, -X SQUARED = X SQUARED.
SO WE HAVE Y = X SQUARED,
WHICH IS EQUIVALENT TO THE ORIGINAL EQUATION.
ALGEBRAICALLY VERIFYING,
THIS EQUATION WOULD HAVE Y AXIS SYMMETRY.
AND THE LAST TYPE OF SYMMETRY WE WANT TO DISCUSS IN THIS VIDEO
IS SYMMETRY ABOUT THE ORIGIN.
AND THE GRAPH OF AN EQUATION IS SYMMETRICAL ABOUT THE ORIGIN
IF WHENEVER THE POINT X, Y IS ON THE GRAPH,
THEN SO IS THE POINT -X, -Y.
SO LOOKING AT THE PROVIDED GRAPH,
IF WE LABELED THIS POINT HERE WITH COORDINATES X, Y,
NOTICE HOW THIS GRAPH WOULD ALSO CONTAIN THIS POINT HERE,
WHICH WOULD HAVE THE COORDINATES -X, -Y.
AND AGAIN, THIS WOULD BE TRUE FOR ANY TWO POINTS
ON THIS GRAPH BECAUSE IT HAS SYMMETRY ABOUT THE ORIGIN.
NOW THE VISUAL TEST
FOR SYMMETRY ABOUT THE ORIGIN IS A LITTLE BIT TRICKIER.
IT'S OFTEN CALLED THE ROTATIONAL TEST,
WHICH MEANS WE ARE GOING TO ROTATE THIS GRAPH
ABOUT THE ORIGIN OR THIS POINT HERE.
AND IF WE CAN ROTATE THIS GRAPH,
ANYTHING LESS THAN ONE COMPLETE TURN
AND IT WOULD LOOK EXACTLY THE SAME,
IT WOULD HAVE SYMMETRY ABOUT THE ORIGIN.
SO FOR EXAMPLE,
IF WE ROTATED THIS GRAPH ONE HALF TURN ABOUT THE ORIGIN,
OR 180 DEGREES, MEANING WE ROTATED IT FROM HERE TO HERE,
ABOUT THE ORIGIN,
NOTICE HOW IT WOULD LOOK EXACTLY THE SAME
BECAUSE THE PIECE IN THE FIRST QUADRANT WOULD BE MOVED
TO THE PIECE IN THE THIRD QUADRANT
AND THEN THE PIECE IN THE THIRD QUADRANT
WOULD BE UP IN THE FIRST QUADRANT
AND THE GRAPH WOULD LOOK EXACTLY THE SAME.
SO THIS IS THE VISUAL TEST FOR SYMMETRY ABOUT THE ORIGIN.
ALGEBRAICALLY,
THE GRAPH OF AN EQUATION WILL HAVE SYMMETRY ABOUT THE ORIGIN
IF WE GET AN EQUIVALENT EQUATION
WHEN X IS REPLACED WITH -X AND Y IS REPLACED WITH -Y.
SO THE EQUATION OF THIS GRAPH IS Y = 1 DIVIDED BY X.
SO NOW TO CHECK FOR SYMMETRY ABOUT THE ORIGIN
WE'RE GOING TO REPLACE Y WITH -Y AND X WITH -X.
WHEN WE DO THIS, AT FIRST, IT WON'T LOOK THE SAME.
WE'D HAVE -Y = 1 DIVIDED BY -X.
AGAIN, IT DOESN'T LOOK THE SAME, BUT IT IS EQUIVALENT.
REMEMBER WE CAN MULTIPLY BOTH SIDES OF AN EQUATION
BY A CONSTANT.
IF WE MULTIPLY BOTH SIDES OF THE EQUATION BY -1,
THIS WOULD JUST CHANGE THE SIGN OF THESE TWO TERMS.
WE WOULD HAVE Y = 1 DIVIDED BY A +X,
WHICH IS EQUIVALENT TO THE ORIGINAL EQUATION.
SO AGAIN, THESE TWO EQUATIONS ARE EQUIVALENT,
AND WE REPLACED X WITH -X AND Y WITH -Y,
THEREFORE WE HAVE SYMMETRY ABOUT THE ORIGIN.
WE'RE GOING TO GO AHEAD AND STOP HERE FOR THIS VIDEO.
THERE IS A LINK ON THE SCREEN TO MORE EXAMPLES OF GIVEN EQUATION,
DETERMINING WHAT TYPES OF SYMMETRY THE GRAPH WOULD HAVE.
NOW THERE'S ONE MORE THING I DO WANT TO MENTION, THOUGH.
MOST GRAPHS DO NOT HAVE SYMMETRY
ABOUT THE X AXIS, Y AXIS, OR THE ORIGIN.
SO KEEP IN MIND THE EXAMPLES THAT WE'RE LOOKING AT
ARE SPECIAL CASES.
OKAY, I HOPE YOU FOUND THIS INTRODUCTORY VIDEO HELPFUL.