- WELCOME TO A SUMMARY OF FUNCTION TRANSFORMATIONS.
THIS VIDEO WILL REVIEW HOW THE VALUES OF "A," B, C AND D
TRANSFORM THE FUNCTION OF F OF X.
LET'S FIRST TALK ABOUT THE VALUE OF "A."
Y = "A" x F OF X.
FOR "A" GREATER THAN 1,
WE'LL STRETCH THE GRAPH OF F OF X VERTICALLY
BY A FACTOR OF "A."
AND Y = "A" x F OF X.
FOR "A" BETWEEN 0 AND 1,
WE'LL COMPRESS THE GRAPH OF F OF X VERTICALLY
BY A FACTOR OF "A."
ONE WAY TO FIND COORDINATES OF A TRANSFORMED FUNCTION
IN THIS FORM WOULD BE TO FIND POINTS ON THE PARENT FUNCTION,
KEEP THE X COORDINATES THE SAME,
BUT MULTIPLY THE Y COORDINATES BY "A"
TO FIND THE Y COORDINATES OF THE TRANSFORMED FUNCTION.
LET'S TAKE A LOOK AT A GRAPH OF THIS.
THE RED FUNCTION IS THE PARENT FUNCTION,
AND THE BLUE FUNCTION WILL BE THE TRANSFORMED FUNCTION.
SO WHEN "A" IS GREATER THAN 1,
THE BLUE FUNCTION WILL BE STRETCHED VERTICALLY
BY A FACTOR OF "A."
ALSO NOTICE THAT THE X INTERCEPTS DO NOT CHANGE
WHEN WE STRETCH THE GRAPH VERTICALLY
OR WHEN "A" IS BETWEEN 0 AND 1 AND THE BLUE FUNCTION
IS COMPRESSED VERTICALLY, AS WE SEE HERE.
NOW, LET'S TALK ABOUT THE VALUE OF B.
WHEN Y = F OF BX AND B IS GREATER THAN 1,
THIS WILL COMPRESS THE GRAPH OF F OF X HORIZONTALLY.
AND WHEN Y = F OF BX AND B IS BETWEEN 0 AND 1,
THIS WILL STRETCH THE GRAPH OF F OF X HORIZONTALLY.
SO AGAIN IF WE KNOW POINTS ON THE PARENT FUNCTION,
WE CAN DETERMINE THE COORDINATES OF THE TRANSFORMED FUNCTION
WITH A DIFFERENT VALUE OF B BY MULTIPLYING EACH X COORDINATE
BY 1/B OR JUST DIVIDING BY B
AND KEEPING THE Y COORDINATES THE SAME.
SO WE CAN SAY THE TRANSFORMED FUNCTION
IS STRETCHED OR COMPRESSED HORIZONTALLY BY A FACTOR OF 1/B.
LET'S GO AHEAD AND TAKE A LOOK AT A GRAPH OF THIS AS WELL.
SO AGAIN, THE TRANSFORMED FUNCTION WILL BE IN BLUE.
AS B IS GREATER THAN 1,
WE'LL SEE THAT THE TRANSFORMED FUNCTION
IS COMPRESSED HORIZONTALLY.
NOTICE THAT THE X INTERCEPTS DO CHANGE NOW.
AND IF B IS BETWEEN 0 AND 1,
THE FUNCTION IS STRETCHED HORIZONTALLY, AS WE SEE HERE.
NOW BEFORE WE TALK ABOUT C AND D,
WE NEED TO BACK UP A LITTLE BIT
AND TALK ABOUT WHAT HAPPENS WHEN WE CHANGE THE SIGN OF A AND B.
LET'S FIRST CONSIDER WHEN "A" = -1.
IF Y = -F OF X, THIS WILL REFLECT F OF X ABOUT THE X AXIS.
REMEMBER F OF X = Y,
SO HERE WE WOULD BE CHANGING THE SIGN OF THE Y COORDINATES
AND THEREFORE REFLECTING THE GRAPH ACROSS THE X AXIS.
IF B IS LESS THAN 1 AND SPECIFICALLY EQUAL TO -1
HERE WHERE Y = F OF -X,
THIS REFLECTS THE ORIGINAL FUNCTION
F OF X ABOUT THE Y AXIS,
BECAUSE WE'RE CHANGING THE SIGN OF THE X COORDINATES
OR THE INPUTS.
LET'S TAKE A LOOK AT THIS AS WELL.
SO WE'LL DECREASE "A."
AND AS SOON AS "A" IS NEGATIVE,
IT WILL BE REFLECTED ACROSS THE X AXIS.
AND WHEN "A" = -1, AS WE SEE HERE,
THE BLUE FUNCTION IS A PERFECT REFLECTION OF THE RED FUNCTION
ACROSS THE X AXIS.
AND WE WILL ALSO HAVE VERTICAL STRETCHES
AND COMPRESSIONS WHEN "A" IS NEGATIVE, AS WE SEE HERE.
NOW, LET'S TAKE A LOOK AT WHAT HAPPENS WHEN B IS EQUAL TO -1.
AS SOON AS B IS NEGATIVE,
IT WILL BE REFLECTED ACROSS THE Y AXIS.
AND THEN WHEN B IS EQUAL TO -1,
IT'S A PERFECT REFLECTION OF THE ORIGINAL FUNCTION
ACROSS THE Y AXIS, AS WE SEE HERE.
AGAIN, WE COULD HAVE HORIZONTAL STRETCHES AND COMPRESSIONS
AS WELL WHEN "A" IS NEGATIVE.
NOW, LET'S TAKE A LOOK AT HOW C AFFECTS THE GRAPH.
IF WE HAVE Y = F OF THE QUANTITY X + C,
THIS WILL SHIFT F OF X LEFT C UNITS.
AND IF WE HAVE Y = F OF THE QUANTITY X - C,
THIS WILL SHIFT THE FUNCTION RIGHT C UNITS.
SO IF WE SEE X + C, IT'LL SHIFT LEFT, AND IF WE SEE X - C,
IT WILL SHIFT RIGHT.
THIS IS ALMOST THE OPPOSITE OF WHAT YOU MIGHT THINK.
LET'S GO BACK TO OUR GRAPH.
NOTICE WHEN WE SEE X - A CONSTANT,
THE BLUE FUNCTION IS SHIFTED RIGHT C UNITS.
AND IF WE SEE F OF X + A CONSTANT,
THE GRAPH IS SHIFTED LEFT C UNITS.
AND LASTLY, WE HAVE D.
IF WE HAVE Y = F OF X + D,
THIS WILL SHIFT THE FUNCTION UP D UNITS.
AND IF WE HAVE Y = F OF X - D,
THIS WILL SHIFT THE FUNCTION DOWN D UNITS.
AND REMEMBER F OF X = Y,
SO THIS IS ADDING AND SUBTRACTING
VALUES TO THE Y COORDINATES.
LET'S TAKE A LOOK AT THIS GRAPHICALLY AS WELL.
IF WE HAVE F OF X + D, THE FUNCTION IS SHIFTED UP D UNITS.
AND IF WE HAVE F OF X - D,
THE FUNCTION IS SHIFTED DOWN D UNITS.
OKAY, LET'S SEE IF WE CAN PUT ALL THESE PIECES TOGETHER.
FOR THESE LAST TWO EXAMPLES,
WE WANT TO BE ABLE TO IDENTIFY THE PARENT FUNCTION
AND THEN DESCRIBE THE TRANSFORMATION.
WE HAVE Y = -2 x THE QUANTITY X - 1 SQUARED + 3.
SO HOPEFULLY WE CAN RECOGNIZE THIS AS A QUADRATIC FUNCTION.
SO THE PARENT FUNCTION WOULD BE Y = X SQUARED.
NEXT, BY LOOKING AT THE FORM OF THE GIVEN FUNCTION,
WE SHOULD BE ABLE TO RECOGNIZE THAT "A" = -2.
SINCE "A" IS NEGATIVE,
THIS WOULD BE A REFLECTION ACROSS THE X AXIS
AND THERE WOULD ALSO BE A VERTICAL STRETCH
BY A FACTOR OF 2.
NEXT, B = 1, SO THERE'S NO AFFECT ON THE GRAPH.
C WOULD BE -1.
REMEMBER C SHIFTS THE GRAPH LEFT AND RIGHT.
BECAUSE IT'S X - 1, THIS WOULD BE A SHIFT RIGHT 1 UNIT.
AND THEN LASTLY, D = + 3.
THIS WOULD PRODUCE A SHIFT UP OF 3 UNITS.
WE'LL STOP HERE ON THIS QUESTION.
THERE ARE TWO OTHER VIDEOS THAT SHOW YOU
HOW TO GRAPH THIS TYPE OF FUNCTION.
IN THIS LAST QUESTION,
WE HAVE Y EQUALS THE SQUARE ROOT OF 1/3 x THE QUANTITY X + 1
AND THEN -3.
SO THE PARENT FUNCTION WOULD BE Y EQUALS THE SQUARE ROOT OF X
OR THE BASIC SQUARE ROOT FUNCTION.
IN THIS EXAMPLE, "A" IS EQUAL TO 1,
SO THERE'S NO AFFECT BY "A."
B = 1/3.
REMEMBER THAT B PRODUCES
A HORIZONTAL STRETCH OR COMPRESSION.
SINCE B IS EQUAL TO 1/3,
THIS WOULD BE A HORIZONTAL STRETCH BY A FACTOR
OF 1/B OR 1/1/3 WHICH IS EQUAL TO 3.
C = 1, THEREFORE WE HAVE A SHIFT LEFT 1 UNIT.
AND THEN LASTLY, D = - 3, SO WE HAVE A SHIFT DOWN 3 UNITS.
OKAY, THAT'LL DO IT FOR THIS VIDEO.
THANK YOU FOR WATCHING.