Question

true or false1. For any quadratic function with real coefficients, = ^2 + + , the x-coordinate of the vertex may be readily calculated by the formula -b/(2a)

Answer 1

Given quadratic equation is

`y=ax^2+bx+c`

The graph of a quadratic function is a parabola with vertex at (h,K).

The general equation of the parabola is

`y=m\cdot(x-h)^2+k`

write the given equation in the form

`y=m(x-h)^2+k`
`\begin{gathered} y=ax^2+bx+c \\ y=a(x^2+(b)/(a)x)+c \\ y=a(x^2+2(x)((b)/(2a))+((b)/(2a))^2)+c-a\cdot((b)/(2a))^2 \\ y=a\cdot(x+(b)/(2a))^2+c-a\cdot((b)/(2a))^2 \\ y=a\cdot(x-(-(b)/(2a)))^2+c-(b^2)/(2a) \end{gathered}`

So, the x- coordinate of the vertex is

`x=-(b)/(2a)`

So, it is true.