- WE WANT TO DETERMINE THE EQUATION
FOR THE EXPONENTIAL FUNCTION FROM THE GRAPH
USING A FORM OF F OF X = "A" x B RAISED TO THE POWER OF X.
BUT THIS ONE'S A LITTLE MORE INVOLVED THAN THE OTHER EXAMPLES
BECAUSE NOTICE HOW THIS GRAPH DOES HAVE A HORIZONTAL ASYMPTOTE
OF Y = +4.
WHICH MEANS THE EXPONENTIAL FUNCTION
WOULD HAVE TO BE SHIFTED UP 4 UNITS
IN ORDER TO HAVE THIS HORIZONTAL ASYMPTOTE.
THIS GRAPH ALSO APPEARS TO BE EXPONENTIAL DECAY
THAT'S BEEN REFLECTED ACROSS THE X AXIS.
SO WE SHOULD RECOGNIZE THAT THE VALUE OF "A"
IS GOING TO BE NEGATIVE,
BUT IF NOTHING ELSE WE DO HAVE TO RECOGNIZE
THAT THE EXPONENTIAL FUNCTION HAS BEEN SHIFTED UP 4 UNITS
BECAUSE OF THIS HORIZONTAL ASYMPTOTE,
WHICH MEANS OUR EXPONENTIAL FUNCTION WILL BE IN THE FORM
OF F OF X = "A" x B RAISED TO THE POWER OF X + 4.
SO ONCE WE RECOGNIZE THIS,
THEN WE CAN DETERMINE THE VALUES OF "A" AND B
BY IDENTIFYING TWO POINTS ON THE FUNCTION
THAT WILL GIVE US TWO FUNCTION VALUES.
SO NOTICE HOW THIS POINT HERE OR THE Y INTERCEPT IS (0, 2)
AND THIS POINT HERE IS THE POINT (-1, -4)
WHICH MEANS IF THE FUNCTION CONTAINS THE POINT (0, 2)
F OF 0 MUST = 2
AND IF THE FUNCTION CONTAINS THE POINT (-1, -4)
THEN F OF -1 MUST = -4.
NOW THAT WE KNOW F OF 0 = 2
WE SHOULD BE ABLE TO USE THIS TO DETERMINE THE VALUE OF "A"
AND THEN WE CAN USE THE SECOND FUNCTION VALUE
TO DETERMINE THE VALUE OF B.
SO IF F OF 0 = 2 WE'LL REPLACE X WITH 0.
SO WE WOULD HAVE "A" x B TO THE 0 + 4 MUST EQUAL +2.
WELL, B TO THE 0 IS EQUAL TO 1
SO WE HAVE "A" + 4 = 2, SUBTRACT 4 ON BOTH SIDES.
SO WE HAVE "A" = -2.
SO NOW WE KNOW THAT F OF X
MUST EQUAL -2 x B TO THE POWER OF X + 4.
NOTICE HOW "A" IS NEGATIVE WHICH IS WHAT WE THOUGHT
BECAUSE IT DOES LOOK LIKE THIS IS EXPONENTIAL DECAY
REFLECTED ACROSS THE X AXIS.
SO NOW WE'LL USE THIS FORM OF OUR FUNCTION
AND OUR SECOND FUNCTION VALUE TO DETERMINE THE VALUE OF B
AND LET'S GO AHEAD AND DO THAT ON THE NEXT SLIDE.
SO WE WERE ABLE TO USE F OF 0 = 2 TO DETERMINE THE VALUE OF "A"
AND NOW WE'LL USE F OF -1 = -4 TO DETERMINE THE VALUE OF B.
SO F OF -1 IS GOING TO BE EQUAL TO -2 x B TO THE -1 + 4
WHICH MUST EQUAL -4.
SO THE FIRST STEP,
LET'S GO AHEAD AND SUBTRACT 4 ON BOTH SIDES.
SO WE HAVE -2 B TO THE -1 = -8.
I'M GOING TO MAKE THIS INTO A FRACTION
AND REMEMBER IF I MOVE THE B DOWN TO THE DENOMINATOR
IT'S GOING TO CHANGE THE SIGN OF THE EXPONENT.
SO WE WOULD HAVE -2/B = -8
AND NOW WE'RE GOING TO MULTIPLY BOTH SIDES OF THE EQUATION BY B.
THIS IS B/1 SO THE Bs SIMPLIFY OUT.
WE HAVE -2 = -8B.
NOW WE CAN DIVIDE BOTH SIDES BY -8.
SO WE HAVE B = THIS SIMPLIFIES TO +1/4.
SO NOW THAT WE HAVE B WE HAVE OUR FUNCTION.
OUR FUNCTION IS F OF X = -2 x THE BASE OF 1/4
RAISED TO THE POWER OF X + 4.
NOW HOPEFULLY WE RECOGNIZE
THAT BECAUSE THE BASE IS BETWEEN 0 AND 1
WE HAVE EXPONENTIAL DECAY.
BECAUSE "A" IS NEGATIVE IT'S REFLECTED ACROSS THE X AXIS.
BECAUSE OF THE 2 IT'S BEEN VERTICALLY STRETCHED
AND BECAUSE OF THE +4 IT'S BEEN SHIFTED UP 4 UNITS
AND THESE TRANSFORMATIONS
DO DESCRIBE THE PROVIDED GRAPH IN RED.
I HOPE YOU FOUND THIS HELPFUL.