solving for a and b given a point where there is a relative minimum

Question

find a and b so that the equation ax^2+bx hasa relative minimum at the point (-1,-7)

Answer 1

First of all, the relative minimum of a parabola means the one and only vertex of that parabola. Therefore, you must have the following:

y = ax² + bx has its vertex (or absolute minimum) at (-1, -7) as given.

y [at x = -1] = -7 = a(-1)² + b(-1) = a - b, so -a + b = 7 [Equation 1]

y [at x = 0] = a(0)² + b(0) = 0 + 0 = 0, so to have a symmetric parabola, we also must have:

y [at x = -2] = a(-2)² + b(-2) = 4a - 2b = 0, so 2a - b = 0 [Equation 2]

Therefore, we can add [Equation 1] and [Equation 2] together to get

[-a + b = 7] + [2a - b = 0] ===> 1a + 0b = a = 7 + 0 = 7, so a = 7,

and then we can plug a = 7 into [Equation 1] to get:

-(7) + b = 7 ===> b = 7 + 7 = 14, so b = 14, and we finally have:

y = ax² + bx = 7x² + 14x.

Answer 2

a parabola with a minimum point is upward opening and the minimum point is the vertex

y-h=a(x-k) where (k,h) = vertex = minimum point = (-1,-7)

y+7 = a(x+1)^2 if the minimum point is (-1,-7). It's a relative and absolute minimum point.

y= ax^2 + 2ax + a-7

a-7 = 0, so a =7

b=2a = 14

y=ax^2 + bx with a=7, b=14

y=7x^2 +14x or

y=7x(x+2)

-7=7(-1)(-1+2) = -7(1) = -7

axis of symmetry is x=-1