Determine a Vertical Stretch or Vertical Compression

Determine a Vertical Stretch or Vertical Compression


– 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.  

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