- WELCOME TO AN INTRODUCTION
TO SYMMETRY ABOUT THE X AXIS, Y AXIS
AND THE ORIGIN USING POINTS.
LET'S BEGIN BY CONSIDERING THESE FOUR POINTS
ON THE COORDINATE PLANE.
MORE SPECIFICALLY, LET'S BEGIN BY CONSIDERING POINT "A"
AND POINT D.
NOTICE HOW IF WE WERE TO FOLD THIS GRAPH
ACROSS THE X AXIS OR THE HORIZONTAL AXIS,
NOTICE HOW POINT "A" AND POINT D WOULD MATCH UP PERFECTLY.
AND, THEREFORE, WE CAN SAY THAT POINT "A" AND POINT D
HAVE X AXIS SYMMETRY, OR ARE SYMMETRICAL
ACROSS THE X AXIS.
IT'S ALSO TRUE IF YOU WERE TO TAKE POINT "A"
AND REFLECT IT ACROSS THE X AXIS,
IT WOULD BE POINT D.
AND IF WE REFLECT POINT D ACROSS THE X AXIS,
IT WOULD BE POINT "A."
SO, ONCE AGAIN, THESE TWO POINTS HAVE SYMMETRY ACROSS THE X AXIS
OR HAVE X AXIS SYMMETRY.
WE CAN ALSO SAY THEY'RE REFLECTIONS
ACROSS THE X AXIS.
BUT LOOK AT THE COORDINATES OF THESE TWO POINTS.
NOTICE HOW THE X COORDINATES ARE THE SAME,
BUT THE Y COORDINATES ARE OPPOSITES.
NOW LET'S CONSIDER POINT "A" AND POINT B.
NOTICE HOW IF WE WERE TO FOLD THIS GRAPH ACROSS THE Y AXIS,
POINT "A" AND POINT B WOULD MATCH UP PERFECTLY
AND, THEREFORE, THESE TWO POINTS ARE SYMMETRICAL ACROSS
THE Y AXIS OR WE SAY THEY HAVE Y AXIS SYMMETRY.
IT'S ALSO TRUE IF WE REFLECT POINT "A" ACROSS THE Y AXIS,
IT WOULD BECOME POINT B.
AND IF WE REFLECT POINT B ACROSS THE Y AXIS,
IT WOULD BECOME POINT "A."
IF WE TAKE A LOOK AT THE COORDINATES
OF THESE TWO POINTS, NOTICE HOW IN THIS CASE
THE Y COORDINATES ARE THE SAME,
BUT THE X COORDINATES ARE OPPOSITES.
THIS IS ALWAYS TRUE WHEN WE HAVE A REFLECTION
ACROSS THE Y AXIS.
NOW LET'S CONSIDER POINT "A" AND POINT C.
THESE TWO POINTS ARE CONSIDERED REFLECTIONS ACROSS THE ORIGIN.
NOTICE IF WE START WITH POINT "A"
REFLECTED ACROSS THE X AXIS FROM HERE TO HERE,
AND THEN REFLECTED AGAIN ACROSS THE Y AXIS, IT WOULD BE POINT C.
SO WHEN WE TAKE A POINT AND REFLECT IT ACROSS
THE X AND Y AXIS, WE SAY IT'S A REFLECTION
ACROSS THE ORIGIN OR THE TWO POINTS
HAVE SYMMETRY ABOUT THE ORIGIN.
ANOTHER WAY TO THINK OF THIS IS, IF YOU WERE TO SKETCH A SEGMENT
WHERE POINT "A" AND POINT C WERE THE END POINTS
AND THEN WE ROTATE THIS 180 DEGREES
ABOUT THE ORIGIN FROM HERE TO HERE,
NOTICE HOW THE TWO POINTS MATCH UP PERFECTLY
WHICH WOULD ALWAYS BE TRUE IF WE HAVE SYMMETRY
ABOUT THE ORIGIN OR IF WE REFLECT
THE TWO POINTS ABOUT THE ORIGIN.
SO WE SAY POINT "A" AND POINT C HAVE SYMMETRY ABOUT THE ORIGIN.
SO TO SUMMARIZE, IF WE HAVE TWO POINTS
AND THE X COORDINATES ARE EQUAL AND THE Y COORDINATES
ARE OPPOSITES, THEN WE HAVE SYMMETRY ABOUT THE X AXIS.
IF THE X COORDINATES ARE OPPOSITES
AND THE Y COORDINATES ARE THE SAME, THEN WE HAVE
SYMMETRY ABOUT THE Y AXIS.
AND FINALLY, IF THE X COORDINATES
ARE OPPOSITES AND THE Y COORDINATES
ARE OPPOSITES, WE HAVE SYMMETRY
ABOUT THE ORIGIN, WHICH MEANS, IF WE'RE GIVEN A SINGLE POINT,
WE WANT TO REFLECT THE POINT ABOUT THE X AXIS TO REPLACE
Y WITH -Y OR CHANGE THE SIGN OF THE Y COORDINATES.
IF WE WANT TO REFLECT THE POINT ABOUT THE Y AXIS,
WE REPLACE X WITH -X OR CHANGE THE SIGN OF THE X COORDINATES.
AND TO REFLECT A POINT ABOUT THE ORIGIN,
WE REPLACE X WITH -X AND Y WITH -Y
WHERE WE CHANGE THE SIGN OF THE X AND Y COORDINATES.
SO LET'S TAKE A LOOK AT AN EXAMPLE.
WE'RE GIVEN A POINT ON THE COORDINATE PLANE
AND ASKED TO FIND THE COORDINATES OF THE POINT
AFTER IT IS REFLECTED ACROSS THE X AXIS, Y AXIS
AND THE ORIGIN.
SO NOTICE THE GIVEN POINT HAS COORDINATES -3, 5.
SO TO VISUALIZE THIS, IF WE REFLECT THIS POINT
ACROSS THE X AXIS OR REFLECTED ACROSS THIS HORIZONTAL AXIS,
THE POINT WOULD GO FROM HERE TO HERE.
NOTICE HOW THE COORDINATES OF THE NEW POINT
WOULD STILL HAVE AN X COORDINATE OF -3, BUT NOW THE Y COORDINATE
WOULD BE -5, WHICH IS THE SAME
AS REPLACING Y WITH -Y OR CHANGING THE SIGN
OF THE Y COORDINATE TO REFLECT THE POINT ACROSS THE X AXIS.
SO THE COORDINATES WOULD BE -3, -5.
TO REFLECT THE GIVEN POINT
ACROSS Y AXIS OR ACROSS THE VERTICAL AXIS,
THE POINT WOULD GO FROM HERE TO HERE.
NOTICE HOW THE COORDINATES OF THE POINT WOULD NOW BE 3, 5.
SO WHEN REFLECTING ACROSS THE Y AXIS,
NOTICE HOW WE CHANGE THE SIGN OF THE X COORDINATE AND LEFT
THE Y COORDINATE THE SAME, WHICH IS WHAT OUR NOTES SAY
REPLACING X WITH -X.
SO FOR THE REFLECTION ACROSS THE Y AXIS,
THE NEW COORDINATES WOULD BE 3, 5.
THEN FINALLY, TO REFLECT ACROSS THE ORIGIN,
WE TAKE THE GIVEN POINT REFLECTED ACROSS THE X AXIS
AND THEN THE Y AXIS, WHICH WOULD BE FROM HERE TO HERE
AND THEN FROM HERE TO HERE.
SO NOTICE HOW THE NEW COORDINATES WOULD BE 3, -5.
COMPARING THIS TO THE DIVISIONAL POINT,
NOTICE HOW WE CHANGE THE SIGN OF THE X COORDINATES
AND THE Y COORDINATES.
SO THE NEW COORDINATES WOULD BE 3, -5.
AND OUR NOTES SAY REPLACE X WITH -X
AND REPLACE Y WITH -Y WHICH WE DID BY CHANGING
THE SIGN OF BOTH COORDINATES.
BUT I DO ALSO WANT TO MENTION THAT,
IF WE REFLECT A POINT ACROSS THE ORIGIN,
THAT WOULD BE THE SAME AS TAKING
THE ORIGINAL BLUE POINT AND ROTATING IT 180 DEGREES
ABOUT THE ORIGIN FROM HERE TO HERE.
I HOPE YOU FOUND THIS HELPFUL.