- WE WANT TO DETERMINE THE EQUATION FOR THE GRAPH IN RED
BASED UPON WHAT WE KNOW ABOUT THE GRAPH OF Y
EQUALS THE ABSOLUTE VALUE OF X WHICH IS GRAPHED IN BLACK.
SO LET'S GO AHEAD AND CALL THE BASIC ABSOLUTE VALUE FUNCTION
F OF X.
SO WE HAVE F OF X EQUALS THE ABSOLUTE VALUE OF X.
LET'S CALL THE RED FUNCTION G OF X.
BECAUSE G OF X IS A TRANSFORMATION OF F OF X,
WE CAN WRITE G OF X IN THIS FORM HERE
WHERE THE VALUES OF A, B, C, AND D
TRANSFORM OUR FUNCTION F OF X.
SO THE FIRST STEP WE SHOULD DESCRIBE
THE TYPE OF TRANSFORMATION THAT HAS OCCURRED
TO FORM THE RED GRAPH.
THE FIRST THING WE SHOULD NOTICE
IS THAT THE V SHAPE OF THE RED GRAPH IS NOT THE SAME
AS THE V SHAPE OF THE BLACK GRAPH.
THE BLACK GRAPH HAS EITHER BEEN VERTICALLY STRETCHED
TO FORM THE RED GRAPH, OR HORIZONTALLY COMPRESSED.
DOESN'T MATTER WHICH TRANSFORMATION WE USE,
BUT IT WILL AFFECT WHETHER WE HAVE TO DETERMINE
THE VALUE OF A OR THE VALUE OF B.
LET'S GO AHEAD AND ASSUME WE HAVE A VERTICAL STRETCH.
WE ALSO SHOULD RECOGNIZE SINCE THE V STILL OPENS UP,
A IS GOING TO BE POSITIVE.
NEXT NOTICE THE LOW POINT OF THE BLACK GRAPH IS THE ORIGIN
BUT THE LOW POINT OF THE RED GRAPH IS THE POINT -3, -2
WHICH MEANS THE RED GRAPH HAS BEEN SHIFTED DOWN 2 UNITS
AND LEFT 3 UNITS.
OKAY SO BECAUSE NOW WE HAVE A VERTICAL STRETCH,
WE NEED TO DETERMINE THE VALUE OF A.
BECAUSE THE GRAPH IS SHIFTED DOWN 2 UNITS,
WE HAVE TO DETERMINE THE VALUE OF D,
AND BECAUSE THE GRAPH IS SHIFTED LEFT 3 UNITS,
WE NEED TO DETERMINE THE VALUE OF C,
WHICH MEANS WE CAN WRITE G OF X IN THE FORM OF G OF X = A
x F OF THE QUANTITY X + C + D.
NOW THERE ARE A COUPLE WAYS OF DETERMINING A,
BUT SINCE WE'RE DEALING WITH THE ABSOLUTE VALUE FUNCTION,
LET'S SEE IF WE CAN ANALYZE THE 2 GRAPHS
TO DETERMINE THE VALUE OF A.
IF WE LOOK AT THE RIGHT SIDE
OF THE BASIC ABSOLUTE VALUE FUNCTION,
NOTICE AS X INCREASES BY 1 UNIT,
THE Y VALUE INCREASES BY 1 UNIT AS WELL.
SO X INCREASES BY 1, Y INCREASES BY 1.
NOW LOOKING AT THE RED GRAPH ON THE RIGHT SIDE,
NOTICE AS X INCREASES BY 1 Y INCREASES BY 2.
AGAIN X INCREASES BY 1 AND Y INCREASES BY 2.
THAT MEANS WE HAVE A VERTICAL STRETCH BY A FACTOR OF 2,
MEANING A = +2.
AGAIN WE KNOW IT'S POSITIVE BECAUSE THE V STILL OPENS UP.
NEXT WE CAN DETERMINE THE VALUE OF C
BECAUSE WE KNOW THE GRAPH HAS BEEN SHIFTED LEFT 3 UNITS.
WHEN THE GRAPH IS SHIFTED LEFT,
THAT MEANS C WILL BE POSITIVE SO C = +3.
REMEMBER IF C WAS NEGATIVE, THE GRAPH WOULD BE SHIFTED RIGHT.
AND THEN BECAUSE THE GRAPH HAS BEEN SHIFTED DOWN TWO UNITS,
WE CAN DETERMINE THE VALUE OF D,
IF D IS NEGATIVE THE GRAPH IS SHIFTED DOWN,
IF D IS POSITIVE THE GRAPH IS SHIFTED UP.
SINCE OUR GRAPH IS SHIFTED DOWN 2 UNITS, D = -2
WHICH MEANS G OF X = 2 x F OF X +3 - 2
AND SINCE F OF X IS THE ABSOLUTE VALUE FUNCTION
WE WOULD HAVE G OF X = 2 x THE ABSOLUTE VALUE OF X + 3 - 2.
SO IF WE'RE ABLE TO ANALYZE THE GRAPH
TO DETERMINE THE VALUE OF A,
WE CAN SEE IT DOES SAVE US SOME WORK
BUT AS I'VE SHOWN ON THE PREVIOUS EXAMPLES,
WE COULD HAVE SELECTED A POINT ON THE RED GRAPH,
THEN PERFORMED SUBSTITUTION TO DETERMINE THE VALUE OF A.
SO REAL QUICKLY, IF WE COULDN'T DETERMINE A
FROM ANALYZING THE GRAPH,
WE COULD SAY THAT G OF X = A x THE ABSOLUTE VALUE OF X + 3 - 2
AND THEN SELECT THE POINT ON THE GRAPH,
LET'S SAY THIS POINT RIGHT HERE WHERE X IS -4 AND Y IS 0,
WHICH MEANS G OF -4 WOULD HAVE TO BE EQUAL TO A x -4 + 3 IS -1.
THE ABSOLUTE VALUE OF -1 - 2 WHICH MUST EQUAL 0.
IF WE SOLVE THIS FOR A WE WOULD HAVE A - 2 = 0.
NOTICE THAT A IS STILL EQUAL TO 2.
SO THERE ARE A COUPLE OF WAYS TO DETERMINE THE VALUE OF A.
I HOPE YOU FOUND THIS HELPFUL.