- WE'RE ASKED TO DETERMINE AN EQUATION FOR A POSSIBLE
TRIG FUNCTION FROM THE TABLE OF VALUES.
SO WHILE IT IS POSSIBLE TO DO THIS JUST BY ANALYZING
THE TABLE OF VALUES, I THINK IT'LL BE HELPFUL
IF WE ACTUALLY PLOT THESE POINTS ON THE COORDINATE PLANE.
SO LET'S GO AHEAD AND DO THAT.
HERE WE HAVE 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 EITHER THE COSINE FUNCTION
OR SINE FUNCTION, MEANING THE BEHAVIOR
OF THESE POINTS RESEMBLES EITHER THE SINE FUNCTION,
AS WE SEE HERE IN BLUE, OR THE COSINE FUNCTION,
AS WE SEE HERE IN RED.
SO IF WE CONNECT THESE POINTS SO THAT IT RESEMBLES EITHER
THE COSINE OR SINE FUNCTION, IT WOULD LOOK
SOMETHING LIKE THIS.
WHETHER WE WANT TO FORM AN EQUATION INVOLVING
THE SINE FUNCTION OR THE COSINE FUNCTION,
WE CAN ACTUALLY USE EITHER ONE.
BUT NOTICE HOW IF WE START AT THE Y-AXIS,
NOTICE HOW WE HAVE A MAXIMUM, AND THEN IT GOES TO THE MIDLINE,
A MINIMUM TO MIDLINE, AND THEN A MAXIMUM.
SO IF WE DON'T WANT TO HAVE A PHASE SHIFT,
WE SHOULD RECOGNIZE THAT THIS RESEMBLES THE COSINE FUNCTION
AS WE SEE HERE IN RED.
SO TO COME UP WITH THE EQUATION FOR THIS FUNCTION,
WE'RE GOING TO FOCUS ON THIS PIECE OF THE GRAPH.
AND NOW THAT WE'VE DECIDED TO USE THE COSINE FUNCTION,
OUR JOB IS TO FIND THE VALUE OF A, B, C, AND D FOR OUR FUNCTION.
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 FUNCTION.
THE VALUE OF C AFFECTS THE VERTICAL TRANSLATION
OR SHIFT UP OR DOWN.
AND THE VALUE OF D AFFECTS THE HORIZONTAL SHIFT
OR PHASE SHIFT, WHICH IS A SHIFT LEFT OR RIGHT.
SO IF WE GO BACK TO OUR FUNCTION,
LET'S START BY IDENTIFYING THE MIDLINE.
THE MIDLINE WILL BE HALFWAY BETWEEN THE MAXIMUM
AND THE MINIMUM.
AND SINCE FROM 4 TO -8 IS 12 UNITS,
AND 12 DIVIDED BY 2 IS 6, OUR MIDLINE WILL BE AT Y = -2.
SO LET'S START BY DETERMINING THE VALUE OF "A," THE AMPLITUDE.
WELL, THE AMPLITUDE IS THE DISTANCE
FROM THE MIDLINE TO THE MAXIMUM OR MIDLINE TO THE MINIMUM.
SO DISTANCE FROM -2 TO 4 IS EQUAL TO 6, THEREFORE A = 6.
WE KNOW IT'S POSITIVE BECAUSE THIS COSINE FUNCTION
HAS NOT BEEN REFLECTED ACROSS THE X-AXIS.
NEXT, THE VALUE OF B IS AFFECTED BY THE PERIOD.
WE CAN SEE ALONG THE X-AXIS, AFTER 8 UNITS,
THE GRAPH STARTS TO REPEAT ITSELF.
SO THE PERIOD IS 8 UNITS, WHICH MEANS
2PI DIVIDED BY B MUST EQUAL 8.
SO WE MULTIPLY BOTH SIDES BY B.
WE HAVE 2PI = 8B.
DIVIDE BOTH SIDES BY 8.
WE HAVE B = PI/4.
SO, AGAIN, SO FAR, WE KNOW A = 6 AND B = PI DIVIDED BY 4.
WELL, THE VALUE OF D AFFECTS THE PHASE SHIFT
OR SHIFT LEFT OR RIGHT.
AND THIS GRAPH HAS NOT BEEN SHIFTED LEFT OR RIGHT,
THEREFORE D = 0.
AND THEN, FINALLY, C AFFECTS THE VERTICAL SHIFT UP OR DOWN.
AND SINCE OUR MIDLINE IS AT Y = -2,
THE GRAPH HAS BEEN SHIFTED DOWN 2 UNITS, THEREFORE C = -2.
AND THIS IS ALL WE NEED TO DETERMINE OUR COSINE FUNCTION
FOR THE TABLE OF VALUES.
OUR FUNCTION WILL BE F(X) = 6 x COSINE PI
DIVIDED BY 4, AGAIN, SINCE D = 0, WE JUST HAVE
TIMES X AND THEN MINUS 2.
THIS WOULD BE A POSSIBLE TRIG FUNCTION BASED UPON
THE TABLE OF VALUES.
IN THE NEXT VIDEO, WE'LL TAKE A LOOK AT THE SAME PROBLEM,
BUT WE'LL SHOW HOW YOU CAN DO IT USING THE SINE FUNCTION.
I HOPE YOU FOUND THIS HELPFUL.