- 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.
