- WELCOME TO TWO EXAMPLES OF DETERMINING
A VERTICAL STRETCH OR VERTICAL COMPRESSION.
G(X) IS A TRANSFORMATION OF F(X) WHERE G(X) = "A" x F(BX).
WE WANT TO DETERMINE THE VALUE OF "A" AND THE VALUE OF B.
TO DO THIS, IT WILL BE HELPFUL TO ANALYZE THE COORDINATES
OF CORRESPONDING POINTS.
SO IN F(X), LET'S FIND THE COORDINATES
OF THESE THREE KEY POINTS.
NOTICE HOW THIS POINT WOULD BE (-6,0).
THIS POINT WOULD BE (2,2), AND THIS POINT WOULD BE (6,-1).
NOW, ON G(X) LET'S FIND THE COORDINATES OF THIS POINT,
THIS POINT AND THIS POINT.
SO THIS POINT HERE IS STILL (-6,0).
THIS POINT HERE THOUGH, IS NOW (2,6),
AND THIS POINT HERE IS NOW (6,-3).
SO LOOKING AT THESE TWO GRAPHS
NOTICE HOW THE X COORDINATES HAVE NOT CHANGED,
BUT THE Y COORDINATES HAVE.
THIS INDICATES WE HAVE A VERTICAL STRETCH
OR VERTICAL COMPRESSION,
BUT NOTICE HOW THE MAXIMUM VALUE HAS INCREASED
AND THE MINIMUM VALUE HAS DECREASED.
THIS INDICATES THAT F(X) HAS BEEN STRETCHED VERTICALLY
TO FORM G(X).
SO WE SHOULD RECOGNIZE THAT MEANS IT IS GOING TO AFFECT
THE VALUE OF "A".
WE DON'T HAVE A HORIZONTAL STRETCH OR COMPRESSION
BECAUSE THE X COORDINATES HAVE NOT CHANGED,
AND THEREFORE B WOULD BE 1.
BUT LET'S GO AHEAD AND TAKE A MOMENT AND REVIEW
HOW "A" AND B AFFECT THE GRAPH OF A BASIC FUNCTION F(X).
LOOKING AT THE VALUE OF "A",
IF "A" IS GREATER THAN 1 THEN WE HAVE A VERTICAL STRETCH
BY A FACTOR OF "A".
NOTICE IF WE LOOK AT Y = F(X) HERE IN BLUE,
Y = 2 x F(X) IS A VERTICAL STRETCH.
AND IF WE GRAPH Y = 0.5 x F(X) WE HAVE A VERTICAL COMPRESSION.
TO FIND POINTS WHEN WE HAVE A VERTICAL STRETCH OR COMPRESSION,
WE MULTIPLY EACH Y COORDINATE BY "A".
NOW, IF WE TAKE A LOOK AT THE VALUE OF B
WHERE IF B IS GREATER THAN 1,
THEN WE HAVE A HORIZONTAL COMPRESSION,
WHICH WE SEE HERE IN GREEN BY THE GRAPH OF Y = F(2X).
THIS IS A HORIZONTAL COMPRESSION OF THE BASIC FUNCTION
Y = F(X).
AND THEN IF B IS BETWEEN ZERO AND ONE,
WE HAVE A HORIZONTAL STRETCH,
AS WE SEE HERE ON THE GRAPH OF Y = F(0.5X).
SO WHEN WE HAVE A HORIZONTAL STRETCH OR COMPRESSION,
WE MULTIPLY THE X COORDINATES OF THE PARENT FUNCTION BY 1
DIVIDED BY B OR 1/B.
THIS WOULD ALSO BE THE SAME AS TAKING THE X COORDINATES
AND DIVIDING BY B.
SO BECAUSE WE ALREADY IDENTIFIED THAT WE HAVE A VERTICAL STRETCH,
WE'LL HAVE TO DETERMINE THE VALUE OF "A",
AND BECAUSE WE DO NOT HAVE
A HORIZONTAL STRETCH OR COMPRESSION, B WOULD BE 1.
SO LOOKING AT THE CORRESPONDING Y COORDINATES,
WE WANT TO DETERMINE--
WE HAVE TO MULTIPLY THE Y COORDINATES OF F(X)
BY TO GET THE Y COORDINATES OF G(X).
SO LET'S LOOK AT THIS SECOND POINT HERE.
NOTICE THAT 2 x 3 = 6 AND -1 x 3 IS ALSO = -3.
AND OF COURSE 0 x 3 IS STILL 0.
SO BECAUSE WE'RE MULTIPLYING EACH Y COORDINATE BY 3
THAT MEANS "A" WOULD BE 3.
SO WE WOULD HAVE G(X) = 3 x F OF--
AGAIN, B IS 1 BECAUSE WE DO NOT HAVE
A HORIZONTAL STRETCH OR COMPRESSION,
SO WE CAN JUST WRITE F(X).
SO "A" WOULD BE = 3 AND B WOULD BE = 1.
LET'S TAKE A LOOK AT A SECOND EXAMPLE.
LET'S BEGIN BY IDENTIFYING THE COORDINATES OF KEY POINTS
ON EACH GRAPH.
SO FOR F(X) WE'LL FIND THE COORDINATES
HERE, HERE, AND HERE.
SO WE HAVE (-7,-4).
HERE WE HAVE (-2,8), AND HERE WE HAVE (6,-6).
AND THEN ON G(X) WE'LL FIND THE COORDINATES
HERE, HERE, AND HERE.
SO WE HAVE (-7,2), AND THEN WE HAVE (-2,-4),
AND THEN HERE WE HAVE (6,3).
WELL, THE FIRST THING YOU MIGHT NOTICE IS THAT THE Y COORDINATES
HAVE ACTUALLY CHANGED SIGNS,
WHICH INDICATES THAT WE HAVE A REFLECTION ACROSS THE X AXIS,
WHICH MEANS THE VALUE OF "A"
WILL BE LESS THAN ZERO OR NEGATIVE.
VISUALLY, WE SHOULD ALSO RECOGNIZE
THAT IN ADDITION TO THE REFLECTION,
F(X) HAS BEEN COMPRESSED VERTICALLY TO FORM G(X).
SO IT'S BEEN VERTICALLY COMPRESSED AND REFLECTED
ACROSS THE X-AXIS.
BUT NOTICE HOW THE X COORDINATES ARE THE SAME,
AND THEREFORE WE DO NOT HAVE
A HORIZONTAL STRETCH OR COMPRESSION,
SO ONCE AGAIN, B WOULD BE = 1.
NOW LET'S ANALYZE THE CORRESPONDING Y COORDINATES
HERE AND HERE.
AGAIN, OUR GOALS ARE TO DETERMINE
WHAT CONSTANT WE HAVE TO MULTIPLY
THESE Y COORDINATES BY TO OBTAIN THESE Y COORDINATES.
SO WE WANT TO DETERMINE WHAT TIMES -4 WOULD BE = 2.
WELL, -4 x -1/2 WOULD BE 2,
AND 8 x -1/2 WOULD BE -4, AND -6 x -1/2 WOULD BE 3,
WHICH MEANS "A" MUST BE -1/2.
SO G(X) = -1/2 x F(X),
AND THEREFORE "A" = -1/2 AND B = 1.
REMEMBER B IS THE COEFFICIENT OF X,
WHICH WOULD BE 1 BECAUSE WE DO NOT HAVE
A HORIZONTAL STRETCH OR COMPRESSION.
I HOPE YOU FOUND THESE TWO EXAMPLES HELPFUL.