- WE'RE GOING TO DETERMINE THE EQUATION FOR THE GRAPH IN RED
BASED UPON WHAT WE KNOW ABOUT THE GRAPH
OF Y = THE ABSOLUTE VALUE OF X WHICH IS GRAPHED IN BLACK.
SO WE'LL LET F(X) BE EQUAL TO
THE BASIC ABSOLUTE VALUE FUNCTION.
WE'LL LET G(X) BE OUR TRANSFORMED FUNCTION
AND THEN BECAUSE G(X) IS A TRANSFORMATION OF F(X),
WE CAN WRITE G(X) IN THE FORM GIVEN HERE BELOW
WHERE "A," B, C AND D TRANSFORM THE GRAPH OF F(X).
SO THE FIRST STEP HERE IS TO DESCRIBE THE TRANSFORMATION
THAT HAS OCCURRED.
AND THERE ARE SEVERAL THINGS GOING ON HERE.
OBVIOUSLY THE V SHAPE IS UPSIDE DOWN
SO IT'S BEEN REFLECTED ACROSS THE X AXIS
BUT THE V IS ALSO NOT THE SAME SHAPE
WHICH MEANS THE BLACK GRAPH
HAS BEEN EITHER VERTICALLY COMPRESSED
TO FORM THE RED GRAPH,
OR HORIZONTALLY STRETCHED.
LET'S GO AHEAD AND ASSUME THAT IT'S BEEN VERTICALLY COMPRESSED
AND AS WE ALREADY SAID, IT'S BEEN REFLECTED
ACROSS THE X AXIS.
THE GOOD NEWS IS THE VALUE OF "A" CONTROLS
BOTH OF THESE TRANSFORMATIONS.
ALSO NOTICE THE V OR THE LOW POINT
ON THE BASIC ABSOLUTE VALUE FUNCTION IS AT THE ORIGIN,
BUT THE HIGH POINT OF THE REFLECTIVE FUNCTION
IS THE POINT HERE AT (1,-2)
WHICH MEANS THE GRAPH HAS BEEN SHIFTED RIGHT 1 UNIT
AND DOWN 2 UNITS.
SO RIGHT 1 UNIT
AND DOWN 2 UNITS.
WELL BECAUSE IT'S RIGHT 1 UNIT,
THAT WILL HELP US DETERMINE THE VALUE OF C
AND BECAUSE IT'S DOWN 2 UNITS,
THAT WILL HELP US DETERMINE THE VALUE OF D,
WHICH MEANS WE CAN WRITE G(X) IN TERMS OF F(X)
AS "A" x F(X + C) + D.
SO WHAT WE'LL DO HERE IS DETERMINE THE VALUES OF C AND D
FROM THE TRANSFORMATIONS
AND THEN USE A POINT ON THE RED GRAPH
TO PERFORM SUBSTITUTION AND SOLVE FOR "A."
BECAUSE THE GRAPH HAS BEEN SHIFTED RIGHT 1 UNIT,
THAT MEANS THE VALUE OF C IS GOING TO BE -1.
REMEMBER IF C IS NEGATIVE,
THE GRAPH IS SHIFTED RIGHT C UNITS
AND IF C IS POSITIVE, THE GRAPH IS SHIFTED LEFT C UNITS.
SO C IS THE OPPOSITE SIGN THAT YOU MIGHT THINK,
BUT IF D IS POSITIVE, THE GRAPH IS SHIFTED UP
AND IF D IS NEGATIVE THE GRAPH IS SHIFTED DOWN.
BECAUSE THIS GRAPH IS SHIFTED DOWN 2 UNITS, D IS -2.
SO THIS GIVES US THE FUNCTION G (X) = "A" x F(X - 1) - 2.
REMEMBER F(X) = THE ABSOLUTE VALUE OF X
SO WE CAN WRITE THIS AS G(X) = "A"
x THE ABSOLUTE VALUE OF X - 1 - 2.
AND NOW TO DETERMINE THE VALUE OF "A,"
WE'LL SELECT ONE POINT ON THE RED GRAPH
AND THEN SOLVE FOR "A."
SO FOR EXAMPLE, IF WE SELECT THIS POINT HERE
WHERE THE X COORDINATE IS -3, AND THE Y COORDINATE IS -4,
THIS MEANS G(-3) MUST EQUAL -4.
LET'S DETERMINE G(-3).
WE WOULD HAVE "A" x THE ABSOLUTE VALUE OF -3 - 1, THAT'S -4 - 2
THIS MUST EQUAL -4.
NOW WE'LL SOLVE FOR "A," THIS WILL BE 4A - 2 = -4.
WE'LL ADD 2 TO BOTH SIDES
THAT GIVES US 4A = -2 DIVIDE BOTH SIDES BY 4
AND WE HAVE "A" = -1/2.
NOTICE "A" IS NEGATIVE, THAT GIVES US A REFLECTION
AND THEN 1/2 GIVES US THE VERTICAL COMPRESSION.
SO NOW WE HAVE G(X),
WE JUST HAVE TO SUBSTITUTE -1/2 FOR "A."
SO WE HAVE G(X) = -1/2 x THE ABSOLUTE VALUE OF X - 1 - 2.
AND THAT WILL DO IT FOR THIS EXAMPLE.
I HOPE THIS WAS HELPFUL.