Question

Find the equation, (f(x) = a(x-h)2+ k), for a parabola containing point (-1,0) and having (-3, 4) as a vertex. What is the standard form of the equation?

Answer 1

$(h;\ k)\ -\ \text{the coordinates of the vertex}$

$\text{therefore we have:}\ f(x)=a(x+3)^2+4$

$\text{a parabola that passes through the point (-1; 0)}$

$\text{substitute the coordinates of the poin (-1; 0) to the equation}$

$(-1;\ 0)\to x=-1;\ y=0\\\\0=a(-1+3)^2+4\\\\0=a(2)^2+4\\\\0=4a+4\ \ \ |-4\\\\4a=-4\ \ \ |:4\\\\a=-1$

$f(x)=-(x+3)^2+4=-(x^2+2\cdot x\cdot3+3^2)+4=-(x^2+6x+9)+4\\\\=-x^2-6x-9+4=-x^2-6x-5$