Given the quadratic function $y = a x^{2} + b x + c$ $y=ax^2+bx+c$ , the maximum value is $a^{2} + 4$ $a^2+ 4$ , at $x = 1$ $x =1$ , and the graph passes through point (3, 1). How do you find the values of the constants a, b, c?

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Answer 1 · 2017-01-12T17:37:02

$(a, b, c) = (- 3, 6, 10), or, (- 1, 2, 4) .$ $(a,b,c)=(-3,6,10), or, (-1,2,4).$

The graph of the function (fun.) $y = a x^{2} + b x + c$ $y=ax^2+bx+c$ passes through

the point (pt.) $(3, 1)$ $(3,1)$ , so, the co-ordinates must satisfy the

equation.

$:. 1=9a+3b+c...................(1)$

$y" is maximum at "x=1 :. [dy/dx]_(x=1)=0 :. 2a+b=0......(2)$

Next,

$y_(max)=a^2+4" occurs at "x=1 :. a+b+c=a^2+4.......(3)$

$(1),(2), &,(3) rArr (a,b,c)=(-3,6,10), or, (-1,2,4).$

For $y_(max)" at x=1, we must have, "[(d^2y)/dx^2]_(x=1) <0$

$dy/dx=2ax+b rArr (d^2y)/dx^2=2a=-6 or -2," readily "<0.$

It is easy to verify that the pair of triads so derived satisfies the given

conditions.

$"Therefore, "(a,b,c)=(-3,6,10), or, (-1,2,4).$

Enjoy Maths.!

Given the quadratic function y=ax^2+bx+cy=ax2+bx+cy=ax^2+bx+c, the maximum value is a^2+ 4a2+4a^2+ 4, at x =1x=1x =1, and the graph passes through point (3, 1). How do you find the values of the constants a, b, c?