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+\frac{b}{a}x)+c \\ y=a(x^2+2(x)(\frac{b}{2a})+(\frac{b}{2a})^2)+c-a\cdot(\frac{b}{2a})^2 \\ y=a\cdot(x+\frac{b}{2a})^2+c-a\cdot(\frac{b}{2a})^2 \\ y=a\cdot(x-(-\frac{b}{2a}))^2+c-\frac{b^2}{2a} \end{gathered}$

So, the x- coordinate of the vertex is

$x=-\frac{b}{2a}$

So, it is true.