Question

What is the vertex form, f(x) = a(x − h)2 + k, for a parabola that passes through the point (4, −3) and has (5, 2) as its vertex. What is the standard form of the equation?

Answer 1

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

$\text{therefore we have:}\ f(x)=a(x-5)^2+2$

$\text{a parabola that passes through the point (4, −3)}$

$\text{substitute the coordinates of the point (4;-3) to the equation:}$

$(4;-3)\to x=4;\ y=-3\\\\-3=a(4-5)^2+2\\\\-3=a(-1)^2+2\\\\-3=a+2\ \ \ |-2\\\\a=-5$

$f(x)=-5(x-5)^2+2=-5(x^2-2\cdot x\cdot5+5^2)+2=-5(x^2-10x+25)+2\\\\=-5x^2+50x-125+2=-5x^2+50x-123$