- WELCOME TO TWO EXAMPLES OF IDENTIFYING
A HORIZONTAL STRETCH OR HORIZONTAL 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'S GOING TO BE HELPFUL
TO ANALYZE THE COORDINATES OF CORRESPONDING POINTS.
SO LET'S FIND THE COORDINATES OF THESE THREE POINTS FOR F(X),
AND THEN THESE THREE POINTS FOR G(X).
SO FOR F(X) THIS POINT HERE HAS COORDINATES (-2,-4),
THIS POINT HAS COORDINATES (1,6),
AND THIS POINT HAS COORDINATES (3,-6).
AND THEN FOR G(X), WE WOULD HAVE (-4,-4), (2,6),
AND (6,-6).
NOW, BY ANALYZING THESE POINTS,
NOTICE HOW THE Y COORDINATES ARE THE SAME,
WHICH MEANS WE DO NOT HAVE VERTICAL STRETCH
OR A VERTICAL COMPRESSION.
BUT THE X COORDINATES HAVE CHANGED,
WHICH MEANS WE HAVE A HORIZONTAL STRETCH OR COMPRESSION.
SO WE SHOULD BE ABLE TO SEE THAT FROM F(X) TO G(X),
F(X) HAS BEEN STRETCHED HORIZONTALLY TO FORM G(X).
SO LOOKING AT OUR EQUATION G(X) = "A" x F(BX),
BECAUSE WE DON'T HAVE A VERTICAL STRETCH OR COMPRESSION
"A" WOULD BE 1.
SO OUR GOAL HERE IS TO FIND THE VALUE OF B,
WHICH AFFECTS THE HORIZONTAL STRETCH OR COMPRESSION.
SO FOR A QUICK REVIEW, AGAIN, IT DOESN'T APPLY IN THIS CASE,
BUT IF WE HAVE Y = "A" x F(X) WHERE "A" IS GREATER THAN 1,
WE HAVE A VERTICAL STRETCH, WHICH WE SEE HERE
BY Y = 2 x F(X).
AND IF "A" IS BETWEEN 0 AND 1, WE HAVE A VERTICAL COMPRESSION,
WHICH WE SEE HERE BY Y = 0.5 x F(X).
BUT, AGAIN, IN OUR CASE, BECAUSE WE HAVE A HORIZONTAL STRETCH,
WE'RE CONCERNED ABOUT FINDING THE VALUE OF B.
SO IN THIS FORM, IF B IS GREATER THAN 1,
WE ACTUALLY HAVE A HORIZONTAL COMPRESSION,
WHICH WE SEE HERE BY Y = F(2X).
AND IF B IS BETWEEN 0 AND 1,
WE HAVE A HORIZONTAL STRETCH, WHICH IS OUR CASE,
WHICH WE SEE BY Y = F(0.5X).
AND THIS SHOULD MAKE SENSE
BECAUSE NOTICE HOW FOR THE SAME FUNCTION VALUES OF Y = F(X),
FOR Y = F(0.5X) IT WOULD TAKE LARGER X VALUES
TO HAVE THE SAME Y VALUES AS F(X).
FOR EXAMPLE, HERE NOTICE HOW IF X WAS 8, 0.5 x 8 WOULD BE 4.
WHEN X IS 8 THIS WOULD BE F(4),
THE SAME AS A FUNCTION VALUE WHEN X IS 4
FOR THE BASIC PARENT FUNCTION.
BUT TO FIND THE VALUE B IS A LITTLE TRICKIER
THAN DETERMINING VALUE OF "A".
IF WE START WITH A PARENT FUNCTION,
WE WOULD MULTIPLY EACH X COORDINATE BY 1/B
TO DETERMINE THE X COORDINATES OF THE HORIZONTALLY STRETCH
OR COMPRESSED FUNCTION.
SO IF WE CAN DETERMINE WHAT WE MULTIPLIED BY
TO FIND THE CORRESPONDING X COORDINATES,
WE WOULD SET THAT VALUE = 1/B AND THEN SOLVE FOR B.
SO GOING BACK TO OUR EXAMPLE,
IF WE TAKE A LOOK AT THE CORRESPONDING X VALUES,
NOTICE HOW -2 x 2 = -4, 1 x 2 = 2, AND 3 x 2 = 6.
THAT DOES NOT MEAN B = 2.
B IS ACTUALLY EQUAL TO THE RECIPROCAL OF 2 OR 1/2.
SO WE COULD SET UP AN EQUATION AS 1 DIVIDED BY B = 2
AND SOLVE FOR B, BUT IT WILL BE THE RECIPROCAL OF 2 OR 1/2.
LET'S GO AND JUST SHOW THAT, PUT THIS OVER 1,
WE GO AND CROSS MULTIPLY 2 x B IS 2B MUST = 1 x 1.
SO 2B = 1, DIVIDE BOTH SIDES BY 2, B = 1/2.
WHICH MEANS G(X) IS EQUAL TO--
AGAIN, "A" IS 1 BECAUSE WE DO NOT HAVE
A VERTICAL STRETCH OR COMPRESSION.
SO WE JUST HAVE F(1/2X).
SO "A" = 1 AND B = 1/2.
LET'S TAKE A LOOK AT A SECOND EXAMPLE.
AGAIN, LET'S FIND THE COORDINATES
OF KEY POINTS ON EACH GRAPH.
SO WE'LL FIND THE COORDINATES HERE, HERE, AND HERE ON F(X).
SO WE HAVE (-6,7), (0,-4), AND (3,3).
AND THEN FOR G(X),
LET'S FIND THE COORDINATES HERE, HERE, AND HERE.
SO WE HAVE (-1,3), (0,-4), AND (2,7).
AND THIS ONE'S A LITTLE TRICKY HERE
BECAUSE NOTICE HOW THE HIGH POINT ON F(X) IS TO THE LEFT,
AND THE HIGH POINT ON G(X) IS TO THE RIGHT,
WHICH MEANS THIS POINT HERE CORRESPONDS TO THIS POINT,
AND THIS POINT HERE CORRESPONDS TO THIS POINT.
SO ONCE AGAIN, NOTICE HOW THE CORRESPONDING POINTS
HAVE THE SAME Y COORDINATES,
WHICH MEANS THERE IS NOT A VERTICAL STRETCH OR COMPRESSION.
BUT THE X COORDINATES HAVE CHANGED,
AND THEY'VE ALSO BEEN REFLECTED
ACROSS THE VERTICAL AXIS OR THE Y-AXIS.
WHICH MEANS F(X) HAS BEEN COMPRESSED HORIZONTALLY
AND ALSO REFLECTED ACROSS THE Y-AXIS.
NOW LET'S COMPARE CORRESPONDING X COORDINATES.
SO COMPARE -6 AND 2, 3 AND -1 AND 0 AND 0.
WE WANT TO DETERMINE WHAT TIMES THESE X COORDINATES
WOULD GIVE US THESE X COORDINATES.
WE CAN DETERMINE THAT VALUE THEN B IS THE RECIPROCAL
OF THAT VALUE.
SO IT MAY BE HELPFUL TO SET UP AN EQUATION IN THIS CASE.
WE WANT TO KNOW -6 x WHAT = 2, SO -6 x X = 2.
OR IF WE WANT, 3 x WHAT WOULD BE -1, 3 x X = -1.
THIS EQUATION'S EASIER TO SOLVE.
DIVIDING BOTH SIDES BY 3 WE HAVE X = -1/3.
OF COURSE, WE COULD SOLVE THIS EQUATION TOO AND STILL GET -1/3.
AGAIN, THIS IS NOT B.
B IS A RECIPROCAL OF THIS VALUE, SO B WOULD BE -3.
BUT IF WE WANTED TO,
WE COULD SET UP AN EQUATION
AND SET 1 DIVIDED BY B = -1/3 PLUS MULTIPLY AND SOLVE FOR B.
SO WE'D HAVE -B = 3 OR B = -3.
REMEMBER THE LESSON ON REFLECTIONS,
IF B IS NEGATIVE WE DO HAVE A REFLECTION ACROSS THE Y-AXIS,
WHICH WE ALREADY NOTICED.
SO WE HAVE G(X) = F(-3X).
SO, AGAIN, "A" IS 1 AND B IS -3.
I HOPE YOU FOUND THIS HELPFUL.