- WE WANT TO DETERMINE
IF EACH TABLE REPRESENTS A LINEAR FUNCTION,
EXPONENTIAL FUNCTION, OR NEITHER.
REMEMBER LINEAR FUNCTION FITS THE FORM F OF X = MX + B.
SO IF X IS INCREASING BY A CONSTANT AMOUNT,
THE FUNCTION VALUES WILL HAVE A COMMON DIFFERENCE.
AN EXPONENTIAL FUNCTION FITS THE FORM F OF X = A x B
RAISED TO THE POWER OF X.
SO IF X INCREASES BY A CONSTANT AMOUNT,
THE FUNCTION VALUES WILL HAVE A COMMON RATIO.
SO LET'S LOOK AT OUR TABLES
AND SEE IF WE CAN RECOGNIZE THE PATTERN.
NOTICE IN ALL THE TABLES X IS INCREASING
BY A CONSTANT AMOUNT,
IN THIS CASE 1.
SO NOW WE'LL ANALYZE THE FUNCTION VALUES.
LOOKING AT THIS FIRST TABLE,
THE FIRST FUNCTION VALUE IS -6, THE SECOND IS -1,
THE THIRD IS 4, AND THE FOURTH IS 9.
IT'S USUALLY EASIEST TO SEE IF A TABLE IS LINEAR,
SO SEE IF THESE FUNCTION VALUES
HAVE A COMMON DIFFERENCE.
SO IF WE TAKE -1 AND SUBTRACT -6, THAT WOULD BE -1 + 6 OR 5,
WHICH IS TELLING US THAT THIS IS LINEAR,
WRITING 5 TO EACH FUNCTION VALUE.
LET'S SEE IF THAT'S TRUE.
-6 + 5 = -1, -1 + 5 = 4, AND 4 + 5 = 9.
THEREFORE ALL THESE FUNCTION VALUES
DO HAVE A COMMON DIFFERENCE OF 5,
WHICH MEANS IT'S A LINEAR FUNCTION.
OF COURSE WE COULD ALSO GRAPH THESE POINTS
TO SEE IF IT'S LINEAR,
WHICH WE'LL DO AT THE VERY END.
LOOKING AT OUR SECOND TABLE,
THE FIRST FUNCTION VALUE IS 50, THE SECOND IS 20,
THE THIRD IS 8, AND THE FOURTH IS 3.2.
TO CHECK TO SEE IF IT'S LINEAR,
WE'LL FIND THE DIFFERENCE
BETWEEN THE FIRST 2 FUNCTION VALUES.
WELL, 20 - 50 = -30, WHICH MEANS IF IT'S LINEAR
THE PATTERN WOULD HAVE TO BE SUBTRACTING 30 EACH TIME.
BUT WE CAN QUICKLY SEE THAT 20 - 30 IS NOT 8,
SO IT CANNOT BE LINEAR.
SO NOW WE'LL TEST TO SEE IF IT'S EXPONENTIAL.
IF IT'S EXPONENTIAL
WE'D BE MULTIPLYING BY THE SAME VALUE EACH TIME,
WHICH IS HARD TO TELL.
AND THAT'S WHY IF IT'S EXPONENTIAL WE CAN TEST TO SEE
IF WE HAVE A COMMON RATIO.
SO NOW WE'LL FIND THE RATIO OF 20 TO 50,
WHICH IS EQUAL TO 2/5 OR 0.4.
SO IF IT'S EXPONENTIAL,
THAT MEANS WE'RE MULTIPLYING BY 0.4 EACH TIME.
WELL 50 x 0.4 = 20, AND 20 x 0.4 = 8, AND 8 x 0.4 = 3.2.
THEREFORE THESE FUNCTION VALUES HAVE THE COMMON RATIO
OF 0.4 OR 2/5,
AND THEREFORE THIS TABLE REPRESENTS
AN EXPONENTIAL FUNCTION.
MOVING ON TO OUR LAST TABLE, WE HAVE A FUNCTION VALUE OF 7,
AND THEN 4, AND THEN 3, AND THEN 4 AGAIN.
TO TEST TO SEE IF IT'S LINEAR, NOTICE HOW TO GO FROM 7 TO 4
WE'D HAVE TO SUBTRACT 3 OR ADD -3.
BUT 4 - 3 DOES NOT EQUAL 3, SO IT CAN'T BE LINEAR.
SO NOW WE'LL TEST TO SEE IF IT'S EXPONENTIAL.
AGAIN, IT'S DIFFICULT TO DETERMINE
WHAT x 7 WOULD GIVE US 4,
SO WE'LL FIND THE RATIO OF 4 TO 7,
WHICH OF COURSE IS JUST 4/7.
SO IF THIS IS EXPONENTIAL,
THEN 4 x 4/7 MUST BE EQUAL TO 3, WHICH IT'S NOT,
WHICH MEANS THIS ALSO CANNOT BE EXPONENTIAL.
SO IF IT'S NOT LINEAR, AND IT'S NOT EXPONENTIAL,
THAT MEANS IT'S NEITHER.
TO DOUBLE-CHECK OUT WORK IT MIGHT BE HELPFUL
TO ALSO LOOK AT THESE POINTS ON THE COORDINATE PLANE.
SO HERE'S OUR FIRST TABLE, WHICH WE SAID WAS LINEAR,
AND NOTICE HOW WE COULD DRAW A LINE
THROUGH THESE FOUR POINTS.
HERE'S THE SECOND TABLE, AND NOTICE HOW IT DOES LOOK
LIKE IT BEHAVES LIKE AN EXPONENTIAL FUNCTION.
IN THIS CASE IT WOULD BE EXPONENTIAL DECAY.
AND THEN FOR OUR LAST TABLE
NOTICE HOW IT'S NOT LINEAR OR EXPONENTIAL,
AND THAT'S WHY IT WAS NEITHER.
OKAY. I HOPE YOU FOUND THIS HELPFUL.