- WE'RE GOING TO DETERMINE AN EQUATION
FOR A POSSIBLE TRIG FUNCTION FROM THE TABLE OF VALUES.
NOW, WE JUST DID THIS PROBLEM, AND WE USED THE COSINE FUNCTION.
NOW WE'LL DO THE SAME PROBLEM AND USE THE SINE FUNCTION.
SO THE FIRST STEP IS GOING TO BE TO PLOT THESE POINTS
ON THE COORDINATE PLANE, SO WE CAN SEE HOW THESE POINTS
ARE BEHAVING.
SO HERE'S THE POINT (0,4),
(2,-2),
(4,-8),
(6,-2),
(8,4),
(10,-2),
AND (12,-8).
SO, RIGHT AWAY, BASED UPON THE BEHAVIOR OF THESE POINTS,
IT SHOULD REMIND US OF THE SINE FUNCTION
OR A COSINE FUNCTION AS WE SEE HERE.
HERE'S THE SINE FUNCTION IN BLUE,
AND HERE'S THE COSINE FUNCTION IN RED.
SO IF WE WERE TO CONNECT THESE POINTS
SO THEY BEHAVE LIKE THE SINE OR COSINE FUNCTION,
IT WOULD LOOK SOMETHING LIKE THIS.
LET'S START BY SKETCHING THE MIDLINE,
WHICH WOULD BE THE HORIZONTAL LINE
HALFWAY BETWEEN THE MAXIMUM OF 4 AND MINIMUM OF -8.
SO THE MIDLINE WOULD BE Y = -2.
AGAIN, IN THE LAST VIDEO, WE NOTICED THAT
SINCE THERE'S A MAXIMUM VALUE ON THE Y-AXIS,
IF WE START HERE AT THE Y-AXIS, THIS PIECE RESEMBLES
THE COSINE FUNCTION.
BUT JUST TO SHOW WE CAN ALSO USE THE SINE FUNCTION,
NOTICE IF WE START IT AT X = 2, THE GRAPH STARTS AT THE MIDLINE.
THEN THE NEXT POINT IS THE MINIMUM,
THE NEXT POINT IS AT THE MIDLINE,
THE NEXT POINT IS AT MAXIMUM, AND THE NEXT POINT
IS BACK TO THE MIDLINE, WHICH MEANS IF WE FOCUS
ON THIS PIECE OF THE GRAPH, WE CAN WRITE A FUNCTION
INVOLVING SINE FOR THE TABLE OF VALUES.
BUT NOTICE IF WE DO THIS, WE WILL HAVE TO SHIFT OUR GRAPH
RIGHT TWO UNITS SO WE BEGIN AT THE MIDLINE.
SO, AGAIN, IF WE TAKE A QUICK LOOK
AT THE SINE FUNCTION HERE IN BLUE,
NOTICE HOW IT STARTS AT THE MIDLINE,
AND THEN A MAXIMUM, MIDLINE, MINIMUM, AND MIDLINE.
SO OUR FUNCTION HAS ALSO BEEN REFLECTED ACROSS THE X-AXIS.
BUT NOW THAT WE KNOW WE WANT TO USE THE SINE FUNCTION
TO MODEL THE TABLE, WE NEED TO FIND
THE VALUE OF A, B, D, AND C.
SO FOR A QUICK REVIEW, FOR BOTH THE SINE AND COSINE FUNCTION,
THE ABSOLUTE VALUE OF "A" IS EQUAL TO THE AMPLITUDE.
2PI DIVIDED BY B IS EQUAL TO THE PERIOD OF THE GRAPH.
THE VALUE OF C AFFECTS THE VERTICAL SHIFT UP OR DOWN.
IF C IS POSITIVE, THE GRAPH IS SHIFTED UP.
IF C IS NEGATIVE, THE GRAPH IS SHIFTED DOWN.
THEN D AFFECTS THE HORIZONTAL TRANSLATION
OR HORIZONTAL SHIFT, SOMETIMES CALLED THE PHASE SHIFT,
WHERE IF WE HAVE X - D, THE SHIFT IS RIGHT.
AND IF WE HAVE X + D, THE SHIFT IS LEFT.
SO GOING BACK TO OUR GRAPH, LET'S SEE IF WE CAN IDENTIFY
THE VALUES OF A, B, D, AND C.
WELL, THE ABSOLUTE VALUE OF "A" IS EQUAL TO THE AMPLITUDE,
WHICH IS THE DISTANCE FROM THE MIDLINE TO THE MAXIMUM
OR THE DISTANCE FROM THE MIDLINE TO THE MINIMUM,
WHICH WE CAN SEE IS SIX UNITS.
BUT BECAUSE THIS IS THE SINE FUNCTION
REFLECTED ACROSS THE X-AXIS, "A" WON'T BE +6,
"A" WILL BE -6.
AGAIN, THIS IS BECAUSE, STARTING AT THE MIDLINE,
THE NEXT POINT IS A MINIMUM.
BUT ON THE BASIC SINE FUNCTION, THE NEXT POINT WOULD BE
A MAXIMUM, WHICH MEANS THE GRAPH IS REFLECTED ACROSS THE X-AXIS.
NEXT, TO DETERMINE THE VALUE OF B,
WE NEED TO RECOGNIZE THAT FROM X = 2 TO X = 10,
THE GRAPH COMPLETES ONE CYCLE AND THEN STARTS TO REPEAT.
THEREFORE THE PERIOD IS EQUAL TO 8, WHICH MEANS
2PI DIVIDED BY B MUST EQUAL 8.
SO WE NEED TO SOLVE THIS EQUATION TO DETERMINE B.
SO WE MULTIPLY BOTH SIDES BY B.
WE WOULD HAVE 2PI = 8B.
DIVIDE BOTH SIDES BY 8, SO WE HAVE B EQUALS,
THIS SIMPLIFIES TO 1PI/4 OR JUST PI/4.
NEXT, LET'S CONSIDER THE VALUE OF D.
REMEMBER, D AFFECTS THE SHIFT LEFT OR RIGHT.
BECAUSE WE'RE USING THE SINE FUNCTION,
THE GRAPH HAS BEEN SHIFTED RIGHT TWO UNITS,
WHICH MEANS THIS MUST BE IN THE FORM OF X - 2
AND THEREFORE D WILL BE 2.
IT'S POSITIVE BECAUSE THE MINUS SIGN
IS ALREADY IN THE FORM OF THE EQUATION.
AND THEN FINALLY THE VALUE OF C AFFECTS THE SHIFT UP OR DOWN.
SINCE THE MIDLINE IS Y = -2,
THE GRAPH IS SHIFTED DOWN 2 UNITS, THEREFORE, C = -2.
NOW THAT WE HAVE THE VALUES OF A, B, D, AND C,
WE CAN WRITE THE EQUATION OF OUR FUNCTION.
THE EQUATION OF OUR FUNCTION USING SINE
WOULD BE F(X) = -6 TIMES SINE OF PI/4
TIMES THE QUANTITY (X-2) - 2.
SO NOW THAT WE'VE DONE THIS USING BOTH THE SINE
AND COSINE FUNCTION, I THINK YOU'LL AGREE
IT'S A LITTLE MORE STRAIGHTFORWARD
USING THE COSINE FUNCTION BECAUSE WE DIDN'T HAVE
A PHASE SHIFT OR A REFLECTION ACROSS THE X-AXIS.
I HOPE YOU FOUND THIS EXPLANATION HELPFUL.