- WELCOME TO ANOTHER VIDEO ON FUNCTION TRANSFORMATIONS.
THIS VIDEO DEALS SPECIFICALLY
WITH HORIZONTAL AND VERTICAL TRANSLATIONS,
SO THE GOAL OF THE VIDEO IS TO GRAPH FUNCTIONS
USING HORIZONTAL AND VERTICAL SHIFTS.
IF WE ARE LOOKING AT TRANSFORMATIONS
FROM THIS FUNCTION TO THIS FUNCTION,
THIS VIDEO DEALS SPECIFICALLY WITH HOW C AND D
TRANSFORM THE FUNCTION.
LET'S FIRST TALK ABOUT A HORIZONTAL SHIFT
WHICH MEANS THE GRAPH WILL BE SHIFTED LEFT OR RIGHT.
Y = F OF THE QUANTITY (X + C),
WE WILL SHIFT F OF X LEFT C UNITS
AND Y = F OF THE QUANTITY X - C.
WE WILL SHIFT F OF X RIGHT C UNITS.
NOW, THIS MAY BE THE OPPOSITE OF WHAT YOU MIGHT THINK.
IF YOU ADD C UNITS TO X, THE FUNCTION WILL BE SHIFTED LEFT,
AND IF YOU SUBTRACT C UNITS FROM X THE SHIFT WILL BE RIGHT.
ONE WAY TO GET A FEEL FOR THIS
WOULD BE TO COMPARE A TABLE OF VALUES FOR F OF X
AND F OF THE QUANTITY X - 1.
SO LET'S GO AHEAD AND DO THAT.
LET'S CHOOSE X = 1, 2, 3, 4.
SO TO FIND Y WE JUST SQUARE X FOR F OF X.
SO THIS WOULD BE 1 SQUARED OR 1, 2 SQUARED WHICH IS 4,
3 SQUARED IS 9, AND 4 SQUARED IS 16.
LET'S USE THE SAME X VALUES FOR F OF X - 1.
SO NOW WE ARE GOING TO SUBTRACT 1 FROM THE INPUT
AND THEN SQUARE IT.
SO 1 - 1 SQUARED WOULD BE 0,
2 - 1 SQUARED WOULD BE 1 SQUARED OR 1.
3 - 1 SQUARED WOULD BE 2 SQUARED OR 4,
AND 4 - 1 SQUARED WOULD BE 3 SQUARED OR 9.
SO IF WE COMPARE THE Y VALUES OF 1, 4, AND 9,
NOTICE THAT FOR F OF X - 1 WE HAVE TO INCREASE X BY 1
IN ORDER TO GET THE SAME Y VALUE.
WHEN WE INCREASE X BY 1
WE ARE SHIFTING THE FUNCTION TO THE RIGHT.
SO WHEN WE SUBTRACT A NUMBER FROM X IT MOVES TO THE RIGHT,
AND WHEN WE ADD A VALUE TO X IT SHIFTS TO THE LEFT.
HERE IS THE GRAPH OF THESE TWO FUNCTIONS,
AND WHAT YOU'LL NOTICE IS FOR ANY CORRESPONDING POINT,
LET'S SAY THE VERTEX ON THE ORIGINAL FUNCTION
AND THE VERTEX ON THE SHIFTED FUNCTION
IT'S ONE UNITS TO THE RIGHT.
PICK ANY POINT ON THE ORIGINAL FUNCTION,
AND THE TRANSLATED FUNCTION WILL BE ONE UNIT TO THE RIGHT.
LET'S LOOK AT AN ANIMATION OF THIS.
HERE WE HAVE AN ORIGINAL FUNCTION IN RED,
AND AS I CHANGE THE VALUE OF C
YOU WILL SEE THE TRANSLATED FUNCTION IN BLUE
AS WELL AS THE FUNCTION NOTATION FOR THE TRANSLATED FUNCTION.
SO NOTICE WHEN IT IS F OF THE QUANTITY X - 2.5
THE BLUE FUNCTION IS SHIFTED TO THE RIGHT 2.5 UNITS.
WE CAN ALSO COMPARE CORRESPONDING POINTS,
AND WHAT YOU WILL FIND IS EACH ONE IN BLUE
IS SHIFTED 2.5 UNITS TO THE RIGHT.
LET'S GO AHEAD AND SEE WHAT HAPPENS
WHEN WE CHANGE THIS TO X + A CONSTANT.
YOU CAN SEE WHEN WE HAVE F OF X + 1
THE TRANSLATED FUNCTION IS SHIFTED TO THE LEFT NOW.
LET'S GO AHEAD AND TALK ABOUT A VERTICAL SHIFT NOW.
Y = F OF X + D WILL SHIFT F OF X UP D UNITS,
Y = F OF X - D WILL SHIFT F OF X DOWN D UNITS,
AND THIS TRANSLATION PROBABLY SEEMS MORE LOGICAL.
REMEMBER F OF X = Y,
SO IF WE ADD D UNITS TO Y THE FUNCTION WILL SHIFT UP.
IF WE SUBTRACT D UNITS FROM Y THE FUNCTION WOULD SHIFT DOWN.
LET'S GO AHEAD AND DO ANOTHER COMPARISON USING F OF X
AND F OF X - 2.
SO AGAIN, FOR F OF X WE'LL JUST SQUARE THE INPUT.
SO WE WILL HAVE 1, 4, 9, 16.
FOR F OF X - 2 WE WILL USE THE SAME INPUTS,
BUT NOW WE'LL SQUARE THE INPUT AND THEN SUBTRACT 2.
1 SQUARED - 2 WOULD BE -1.
2 SQUARED - 2, THAT WOULD BE 4 - 2 OR 2.
3 SQUARED - 2, THAT WOULD BE 9 - 2 OR 7,
AND 4 SQUARED - 2 WOULD BE 16 - 2 OR 14.
SO IF WE DO ANOTHER COMPARISON OF THE Y-VALUES OF THE FUNCTION,
NOTICE THAT ALL OF THE Y VALUES IN RED ARE 2
LESS THAN THE Y-VALUES IN BLUE,
THEREFORE THIS FUNCTION WOULD BE 2 UNITS LOWER
THAN THE ORIGINAL.
HERE IS THE GRAPH OF THOSE 2 FUNCTIONS,
AND SO WE CAN PICK ANY POINT ON THE ORIGINAL BLACK FUNCTION.
TO FIND THE CORRESPONDING POINT ON THE TRANSLATED FUNCTION
WE WOULD JUST MOVE THIS POINT DOWN 2 UNITS.
LET'S GO AHEAD AND TAKE A LOOK AT AN ANIMATION OF THIS AS WELL.
SO AS WE CHANGE THE VALUE OF D
WE WILL SEE HOW IT AFFECTS THE GRAPH.
THE TRANSLATED GRAPH WILL BE IN BLUE.
AS WE INCREASE D, THE FUNCTION IS SHIFTED UPWARD.
IF WE HAVE F OF X - D
THE FUNCTION IS SHIFTED DOWN FROM THE ORIGINAL.
LET'S GO AHEAD AND TAKE A LOOK AT SOME EXAMPLES.
WE WANT TO BE ABLE TO USE WHAT WE JUST LEARNED
IN ORDER TO ACCURATELY AND QUICKLY GRAPH F OF X
= THE ABSOLUTE VALUE OF THE QUANTITY X + 3 + 2.
THE FIRST THING WE NEED TO DO
IS RECOGNIZE WHAT THE PARENT FUNCTION IS,
AND IN THIS CASE IT WOULD BE THE ABSOLUTE VALUE OF X.
LET'S GO AHEAD AND CALL IT G OF X = THE ABSOLUTE VALUE OF X.
LET'S GO AHEAD AND GRAPH THIS FOR REFERENCE.
REMEMBER THE ABSOLUTE VALUE FUNCTION FORMS A V,
LOOKS SOMETHING LIKE THIS.
LET'S GO AHEAD AND IDENTIFY A FEW OF THESE POINTS.
THIS WOULD BE (2,2).
THIS WOULD BE (0,0), AND THIS WOULD BE (-2,2).
THE NEXT THING WE NEED TO BE ABLE TO DO
IS RECOGNIZE HOW TAKING THE ABSOLUTE VALUE OF X + 3
AND THEN ADDING 2 WOULD TRANSLATE THE PARENT FUNCTION.
SO WHAT WE COULD DO IS SAY THAT F OF X = G OF X + 3 + 2
IF THAT'S HELPFUL.
NOTICE THAT G IS JUST THE ABSOLUTE VALUE OF X,
G OF X + 3 WOULD BE THIS PART OF THE FUNCTION,
AND THEN THE +2 WOULD BE THE CONSTANT ON THE END.
INCREASING X BY 3 HERE
WHICH WOULD BE THE SAME AS THIS X + 3 HERE
WOULD SHIFT THE GRAPH LEFT 3 UNITS.
THEN ADDING 2 TO THE ABSOLUTE VALUE FUNCTION HERE OR HERE
WOULD SHIFT THE FUNCTION UP 2 UNITS.
SO NOW WHAT WE CAN DO IS TAKE THESE 3 KEY POINTS
AND SHIFT THEM LEFT 3 UNITS AND UP 2
TO GRAPH THE GIVEN FUNCTION.
LET'S DO THAT.
LET'S START WITH THE LEFTMOST POINT.
WE'RE GOING TO GO LEFT 3 UNITS AND THEN UP 2,
SO WE'D BE OVER TO -5 AND THEN UP TO 4.
NEXT, WE TAKE THIS POINT (0,0)
AND SHIFT IT LEFT 3 UNITS AND UP 2.
WE WOULD BE HERE AT (-3,2).
AND LASTLY, WE'LL TAKE THIS RIGHTMOST POINT,
SHIFT IT LEFT 3 UNITS AND UP 2,
AND WE'D BE RIGHT HERE AT (-1,4).
NOW THAT WE KNOW THAT THE ABSOLUTE VALUE FUNCTION
IS A V SHAPE
WE CAN FORM THE NEW FUNCTION HERE IN GREEN
USING TRANSLATIONS,
AND IT WOULD LOOK SOMETHING LIKE THAT.
LET'S GO AHEAD AND TRY ANOTHER ONE.
AGAIN, THE FIRST STEP IS GOING TO BE TO RECOGNIZE
WHAT THE PARENT FUNCTION WOULD BE.
IF F OF X = THE QUANTITY X - 2 SQUARED - 4
WE SHOULD RECOGNIZE THE PARENT FUNCTION
AS, LET'S CALL IT G OF X = X SQUARED.
SO IF WE WANT TO CREATE F OF X USING G OF X,
F OF X IS GOING TO EQUAL G OF--
WELL, X - 2 IS BEING SQUARED,
SO WE'D HAVE G OF X - 2, AND THEN WE'RE SUBTRACTING 4.
NOW AFTER ALL, YOU MAY NOT HAVE TO WRITE IT LIKE THIS,
BUT IT MAY HELP AT THE BEGINNING TO RECOGNIZE
THAT HERE WE ARE DECREASING THE INPUT OF X BY 2
WHICH MEANS IT WILL SHIFT IT RIGHT 2 UNITS.
RECOGNIZING THIS X - 2 HERE IS THE SAME AS THE X - 2 HERE,
AND THEN SUBTRACTING 4 FROM THIS FUNCTION VALUE
WOULD SHIFT IT DOWN 4 UNITS.
SO THIS - 4 IS THE SAME AS THIS - 4 HERE IN THE GIVEN FUNCTION.
SO TO GRAPH THIS GIVEN FUNCTION
WE SHOULD FIRST SKETCH THE GRAPH OF THE PARENT FUNCTION
G OF X = X SQUARED.
LET'S GO AHEAD AND DO THAT.
REMEMBER, THAT IS A PARABOLA WITH ITS VERTEX AT THE ORIGIN.
SO THIS WOULD BE THE VERTEX. 1 SQUARED WOULD BE 1.
2 SQUARED WOULD BE 4,
AND THEN WE HAVE A MIRROR IMAGE ON THE OTHER SIDE OF THE Y-AXIS.
SO WE HAVE A POINT HERE AND A POINT HERE.
SO THIS IS THE PARENT FUNCTION,
AND WHAT WE'RE GOING TO DO NOW IS PICK SOME KEY POINTS HERE
AND THEN SHIFT EACH OF THEM RIGHT 2 UNITS AND DOWN 4 UNITS.
SO LET'S START WITH THE VERTEX.
WE'LL SHIFT IT RIGHT 2 UNITS AND DOWN 4 UNITS TO HERE.
NEXT, WE WILL TAKE THIS POINT HERE
AND SHIFT IT RIGHT 2 UNITS AND DOWN 4 UNITS.
THAT WOULD BE HERE AT (1,-3).
THEN WE'LL TAKE THE 0.11, SHIFT IT RIGHT 2 AND DOWN 4 HERE.
MIGHT AS WELL GO AHEAD AND DO THESE LAST TWO POINTS.
LET'S TAKE THIS ONE AND SHIFT IT TO THE RIGHT 2, DOWN 4.
IT WOULD BE HERE ON THE X-AXIS.
AND THE SAME HERE, SHIFT IT RIGHT 2 AND DOWN 4.
WE WOULD BE AT THE ORIGIN NOW.
SO THE TRANSLATED FUNCTION WOULD BE THIS GREEN PARABOLA
AS WE SEE HERE.
THANK YOU FOR WATCHING.