- WELCOME TO ANOTHER VIDEO ON FUNCTION TRANSFORMATIONS.
THIS VIDEO DEALS SPECIFICALLY WITH THE REFLECTION
ABOUT THE X-AXIS AND THE Y-AXIS.
SO IF WE'RE COMPARING F(X) TO THE FUNCTION IN THIS FORM,
WE'LL BE TAKING A LOOK AT HOW THE SIGN OF "A"
AND THE SIGN OF B AFFECT THE GRAPH OF THE ORIGINAL FUNCTION.
SO EVEN THOUGH WE HAVE TALKED
ABOUT VERTICAL AND HORIZONTAL STRETCHES AND COMPRESSIONS,
WE HAVEN'T TALKED ABOUT WHAT HAPPENS WHEN WE CHANGE THE SIGN
OF "A" AND B.
Y = -F(X) REFLECTS F(X) ABOUT THE X-AXIS.
REMEMBER F(X) = Y,
SO IF WE CHANGE THE SIGN OF THE Y COORDINATES
IT WOULD REFLECT THE GRAPH ABOUT THE X-AXIS.
IF Y = F(-X), THIS WOULD REFLECT F(X)
ABOUT THE Y-AXIS AND HERE WE'RE CHANGING THE SIGN
OF THE X COORDINATES AND THEREFORE WOULD REFLECT
THE GRAPH ACROSS THE Y-AXIS.
LET'S GO AHEAD AND TAKE A LOOK AT THIS USING T TABLES
FOR COMPARISON.
IF F(X) = THE SQUARE ROOT OF X, THEN TO DETERMINE Y,
WE WOULD TAKE THE SQUARE ROOT OF THESE X VALUES.
SO THE SQUARE ROOT OF 0 = 0, SQUARE ROOT OF 1 = 1,
SQUARE ROOT OF 4 = 2, SQUARE ROOT OF 9 = 3.
NOW LET'S COMPARE THIS TO G(X) = -F(X) = -SQUARE ROOT X.
SO THE SQUARE ROOT OF X WOULD STILL BE THE SAME EACH TIME,
BUT NOW WE WOULD TAKE THE OPPOSITE OF THAT
SO WE WOULD HAVE 0, -1, -2, AND -3.
AND NOTICE THAT IF YOU CHANGE THE SIGN OF THE Y COORDINATES,
THIS WOULD REFLECT THE GRAPH ACROSS THE X AXIS.
NOW, IF WE LET H(X) = F(-X),
THIS WOULD EQUAL THE SQUARE ROOT OF -X.
REMEMBER THIS RADICAND MUST BE GREATER THAN OR EQUAL TO ZERO
TO TAKE THE SQUARE ROOT OF IT.
SO IF WE CHANGE THE SIGN OF THE X COORDINATES--
SO IF WE CHANGE THE SIGN OF OUR INPUT X,
IT WOULD MAKE OUR RADICAND EITHER POSITIVE OR ZERO.
SO WE'D HAVE 0, 1, 2, AND 3.
SO SINCE WE CHANGED THE SIGN OF THE X COORDINATES
IT WOULD REFLECT IT ACROSS THE Y-AXIS.
IF WE WERE TO GRAPH THESE THREE
FUNCTIONS ON THE SAME COORDINATE PLANE,
WE WOULD HAVE OUR ORIGINAL FUNCTION IN BLUE.
IF WE CHANGE THE SIGN OF THE X COORDINATES
OR REFLECT THIS GRAPH ABOUT THE Y-AXIS,
WE WOULD HAVE H(X) = F(-X).
WE'D CHANGE THE SIGN OF THE X COORDINATE.
AND IN GREEN, IF WE REFLECT IT ACROSS THE X AXIS,
OR CHANGE THE SIGN OF THE Y COORDINATES,
WE WOULD HAVE G(X) = -F(X).
LET'S TAKE A LOOK AT AN ANIMATION OF THIS.
SO WHAT WE SEE HERE IS THE ORIGINAL FUNCTION IN RED
AND WE'LL SEE THE TRANSFORMED FUNCTION IN BLUE.
WE'LL START BY CHANGING THE VALUE "A" FROM +1 TO -1.
HERE YOU ARE SEEING A VERTICAL COMPRESSION.
AND THEN AS SOON AS "A" TURNS NEGATIVE,
WE'LL SEE A REFLECTION ACROSS THE X-AXIS.
AND AS SOON AS "A" = -1 THE TRANSFORMED FUNCTION IN BLUE
IS A PERFECT REFLECTION OF THE ORIGINAL FUNCTION IN RED
ACROSS THE X-AXIS.
LETS NOW TAKE A LOOK AT WHAT HAPPENS WHEN B CHANGES
FROM 1 TO -1.
SO RIGHT NOW B = +1 AND AS WE DECREASE B,
YOU'LL SEE A HORIZONTAL STRETCH.
AND THEN WHEN B TURNS NEGATIVE,
WE'LL SEE A REFLECTION ACROSS THE Y-AXIS.
AND WHEN B = -1, WE HAVE A PERFECT REFLECTION
OF A RED FUNCTION ACROSS THE Y-AXIS.
LET'S GO AHEAD AND TAKE A LOOK AT SOME OF OUR OWN EXAMPLES NOW.
AGAIN, THE FIRST THING WE WANT TO BE ABLE TO RECOGNIZE
IS THAT OUR PARENT FUNCTION
WOULD BE G(X) = THE ABSOLUTE VALUE OF X,
WHICH WE SHOULD KNOW BY NOW IS JUST A V THAT OPENS UPWARD.
AND SOME OF THE KEY POINTS ON THIS GRAPH
WOULD BE (0,0), (3,3), AND THEN MAYBE (-3,3).
SO F(X) IS REALLY JUST THE OPPOSITE OF G(X),
WHICH IS EQUAL TO NEGATIVE ABSOLUTE VALUE OF X.
AND HERE WE SHOULD BE ABLE TO RECOGNIZE THAT "A" = -1
OR THAT WE'RE CHANGING THE SIGN OF THE Y COORDINATES,
THEREFORE IT'S A REFLECTION ACROSS THE X-AXIS.
SO INSTEAD OF OPENING UP, THE GIVEN FUNCTION WILL OPEN DOWN.
SO WE CAN JUST CHANGE THE SIGN OF THESE Y COORDINATES,
SO WE'D HAVE THE POINT (-3,-3),
SO THE ORIGIN IS THEN (3,-3).
THIS WOULD BE THE GRAPH F(X) = NEGATIVE ABSOLUTE VALUE OF X.
AGAIN, THE ORIGINAL IS IN RED
AND THE TRANSFORMED, OR REFLECTED GRAPH IS IN BLUE.
AND HERE WE HAVE F(X) = THE CUBED ROOT OF -X,
SO OUR PARENT FUNCTION WOULD BE THE CUBED ROOT OF X.
NOW THIS ONE MAY NOT BE QUITE
AS OBVIOUS AS SOME OF THE OTHERS THAT WE'VE DONE.
LET'S GO AHEAD AND SKETCH THIS PARENT GRAPH.
SO IF WE TAKE THE CUBE ROOT OF -8 THAT WOULD BE -2,
SO WE'LL PLOT THE POINT (-8,-2), (0,0),
AND THEN THE CUBED ROOT OF 8 WOULD BE 2.
LET'S FIND TWO MORE POINTS.
THE CUBED ROOT OF -27 WOULD BE -3, SOMEWHERE HERE.
AND THE CUBED ROOT OF +27 IS 3.
SO THE PARENT FUNCTION LOOKS SOMETHING LIKE THIS,
BUT WE WANT TO GRAPH F(X) = THE CUBED ROOT OF -X.
SO F(X) = G(-X), WHICH = THE CUBE ROOT OF -X.
SO HERE NOTICE WE'RE CHANGING THE SIGN OF THE INPUT
OR EVERY X-COORDINATE.
SO LET'S GO AHEAD AND JUST RE-PLOT THESE POINTS BY CHANGING
THE X COORDINATES.
SO WE STILL HAVE (0,0) AND THEN INSTEAD OF (8,2)
WE'LL HAVE (-8,2).
INSTEAD OF (27,3), WE'LL HAVE (-27,3),
AND INSTEAD OF (-8,-2) WE'LL HAVE (+8,-2),
AND LASTLY, INSTEAD OF (-27,-3) WE'LL HAVE (+27,-3).
SO OUR TRANSFORMED GRAPH IS GREEN
AND WOULD LOOK SOMETHING LIKE THIS
AND NOW WE CAN SEE IT'S A REFLECTION ABOUT THE Y-AXIS.
OKAY, IN THE NEXT TWO VIDEOS WHAT WE'RE GOING TO DO
IS TAKE A LOOK AT VARIOUS TRANSFORMATIONS ALL COMBINED
AT THE SAME TIME TO FINISH OFF
THE TOPIC OF FUNCTION TRANSFORMATIONS.
THANK YOU FOR WATCHING.