Ex - Find a quadratic function given the vertex and the leading coefficient

IN THIS PROBLEM WE'RE ASKED TO DETERMINE THE VALUE OF B AND C FOR OUR QUADRATIC FUNCTION THAT HAS A VERTEX WITH AN X COORDINATE OF 3 AND A Y COORDINATE OF -17. SO LOOKING AT OUR FUNCTION AGAIN, NOTICE THAT "A", THE LEADING COEFFICIENT, = -2, AND THEN WE'RE ACTUALLY DETERMINING THE VALUE OF B AND THE VALUE OF C. IF WE LOOK AT THE GRAPH OF A QUADRATIC FUNCTION JUST FOR A MOMENT, WE SHOULD KNOW THE EQUATION FOR THE AXIS OF SYMMETRY IS X = -B DIVIDED BY 2A, WHEN OUR FUNCTION IS WRITTEN IN GENERAL FORM HERE. WELL, THE VERTEX, OR THIS POINT HERE, IS THE ONLY POINT THAT'S ON THE PARABOLA, OR THE GRAPH OF OUR FUNCTION, AND THE AXIS OF SYMMETRY. THEREFORE THE X COORDINATE OF THE VERTEX = -B DIVIDED BY 2A, THE SAME FORMULA USED TO DETERMINE THE EQUATION OF THE AXIS OF SYMMETRY. AND THEN THE Y COORDINATE OF THE VERTEX, BECAUSE IT'S ON THE FUNCTION, WOULD HAVE TO BE F OF -B DIVIDED BY 2A. WE CAN USE THIS INFORMATION TO SOLVE THIS GIVEN PROBLEM. SO IF WE START WITH THE EQUATION OF THE AXIS OF SYMMETRY, IT ALSO GIVES US THE X COORDINATE OF THE VERTEX. SO BECAUSE THE X COORDINATE OF OUR VERTEX IS 3, THIS X MUST = 3, AND THIS MUST = -B DIVIDED 2 x "A," AND WE KNOW "A" IS -2. SO HERE WE WOULD HAVE 3 = -B DIVIDED BY -4. WELL, A NEGATIVE DIVIDED BY A NEGATIVE IS POSITIVE, SO WE HAVE B DIVIDED BY 4. AND NOW MULTIPLYING BOTH SIDES BY 4 WE CAN SEE THAT B = 12. LET'S GO AHEAD AND RECORD THIS. WE KNOW B = 12. NOW, GOING BACK TO OUR VERTEX NOTICE WHEN X = 3, Y = -17, WHICH MEANS IF WE EVALUATE OUR FUNCTION, OR DETERMINE F OF 3, IT MUST = -17. SO F OF 3 = -2 x 3 SQUARED. WE KNOW B IS 12, SO WE'D HAVE + 12 x 3 + C, AND WE KNOW THIS MUST EQUAL -17. AND NOW WE'LL SOLVE THIS FOR C. SO 3 SQUARED IS 9 x -2 = -18 + 12 x 3 = 36 + C = -17. COMBINING LIKE TERMS THIS WOULD BE 18 + C = -17, AND FINALLY SUBTRACTING 18 ON BOTH SIDES, WE HAVE C = -35. SO IF C = -35 WE NOW KNOW OUR QUADRATIC FUNCTION. IT IS F OF X = -2X SQUARED + 12X - 35. OKAY. HOPE YOU FOUND THIS EXPLANATION HELPFUL.