- WE WANT TO DETERMINE THE EQUATION
OF THE QUADRATIC FUNCTION FROM THE GRAPH.
WE ALSO WANT TO GIVE THE FUNCTION IN GENERAL FORM.
REMEMBER GENERAL FORM OF A QUADRATIC FUNCTION
IS F OF X = AX/SQUARED + BX + C.
BUT TO FIND THIS FORM OF THE FUNCTION,
WE'RE ACTUALLY GOING TO USE THE STANDARD FORM FIRST,
WHICH IS F OF X = "A" x THE QUANTITY X - H/SQUARED + K.
SO THE FIRST THING WE'LL DO IS REPLACE F OF X WITH Y
SO WE'LL HAVE Y = "A" x THE QUANTITY X - H/SQUARED + K.
SO IN THIS FORM, HK WOULD BE THE COORDINATES OF THE VERTEX
AND XY WOULD REPRESENT SOME POINT ON THE GRAPH.
SO NOW LET'S TAKE A LOOK AT OUR GRAPH.
THIS WOULD BE OUR VERTEX.
NOTICE HOW THE VERTEX WOULD HAVE AN X COORDINATE OF -3
AND A Y COORDINATE OF +4.
AND THEN NOTICE THIS POINT HERE ON THE FUNCTION
WOULD HAVE AN X COORDINATE OF -4 AND A Y COORDINATE OF +2.
THIS IS ACTUALLY ALL THE INFORMATION WE NEED
TO GET STARTED.
SINCE THIS POINT IS THE VERTEX, H = -3 AND K = +4
AND SINCE THIS POINT IS ON THE FUNCTION
WE CAN USE X = -4 AND Y = +2.
SO AGAIN FROM HERE WE'RE GOING TO SUBSTITUTE -3 FOR H,
+4 FOR K, -4 FOR X, AND +2 FOR Y.
NOTICE WHEN WE DO THIS WE'LL BE LEFT WITH ONE UNKNOWN, "A".
ONCE WE DETERMINE THE VALUE OF "A"
WE'LL HAVE THE FUNCTION IN STANDARD FORM
WHICH WE'LL THEN WRITE IN GENERAL FORM.
SO WERE GOING TO HAVE 2 = "A" x THE QUANTITY X - H,
WELL X IS -4 - H IS GOING TO BE - -3.
AND WE'RE GOING TO SQUARE THIS AND THEN ADD K AND K IS +4.
SO NOW LET'S GO AHEAD AND SIMPLIFY THIS.
WE'LL HAVE 2 = "A" x THIS IS GOING TO BE -4 + 3
OR -1/SQUARED + 4.
SO WE'LL HAVE 2 = -1/SQUARED IS JUST 1
SO 1 x "A" IS "A" + 4.
SUBTRACT 4 ON BOTH SIDES OF THE EQUATION.
THIS IS 0. SO WE HAVE -2 = "A".
SO OUR FUNCTION IN STANDARD FORM
IS GOING TO BE F OF X = "A" WHICH IS -2
AND THEN WE'LL HAVE THE QUANTITY X - H/SQUARED.
REMEMBER WE DO WANT X IN OUR FUNCTION
SO WE'RE GOING TO HAVE X - H, BUT H IS -3, - -3 WILL BE +3.
SO WE HAVE X - 3/SQUARED AND THEN + K.
AGAIN K IS 4, SO WE HAVE + 4.
BUT SINCE WE WANT THIS IN GENERAL FORM
WE'LL HAVE TO MULTIPLY THIS OUT, COMBINE LIKE TERMS
AND THEN PUT THE TERMS IN DESCENDING ORDER
SO IT'S IN GENERAL FORM.
SO LET'S DO THAT ON THE NEXT SLIDE.
SO WE'LL START BY SQUARING THE QUANTITY X + 3.
REMEMBER THERE ARE NO SHORTCUTS.
WE'LL HAVE TO WRITE THAT OUT AS TWO FACTORS
AND THEN MULTIPLY THE TWO BINOMIALS.
SO WE'LL HAVE FOUR PRODUCTS.
1, 2, 3, AND 4.
SO I HAVE -2 x THIS PRODUCT WHICH WILL BE X/SQUARED.
THEN WE'LL HAVE + 3X + 3X, THAT'S + 6X + 9 + 4.
NOW WE'LL DISTRIBUTE THE -2.
SO WE'LL HAVE -2X/SQUARED - 12X - 18 + 4
AND WE HAVE TWO LIKE TERMS HERE THAT WE CAN COMBINE.
SO THE FUNCTION IN GENERAL FORM WOULD BE
F OF X = -2X/SQUARED - 12X
AND THEN -18 + 4 IS GOING TO BE -14 SO WE HAVE - 14.
SO THIS WOULD BE THE GENERAL FORM OF OUR QUADRATIC FUNCTION
GIVEN BY THIS GRAPH.
I HOPE YOU FOUND THIS HELPFUL.