How do you find the axis of symmetry, and the maximum or minimum value of the function y = 3x^2 + 24x - 1?

May 30, 2016

$3 {x}^{2} + 24 x - 1 \equiv 3 {\left(x + 4\right)}^{2} - 49$

Explanation:

Giving a parabola written as

$y = a {\left(x - {x}_{0}\right)}^{2} + b$

geometrically speaking we can qualify:

{(x_0 = "axis of symmetry"), (a = "scale factor"), (b = "offset value"):}

Given a parabola such as

$y = 3 {x}^{2} + 24 x - 1$

we can reduce it to the former formulation, making

$3 {x}^{2} + 24 x - 1 = a {\left(x - {x}_{0}\right)}^{2} + b$

and equating the coefficients, results in

$\left\{\begin{matrix}- 1 - b - a {x}_{0}^{2} = 0 \\ 24 + 2 a {x}_{0} = 0 \\ 3 - a = 0\end{matrix}\right.$

solving for $a , b , {x}_{0}$

$\left\{a = 3 , b = - 49 , {x}_{0} = - 4\right\}$

and the equivalent formulation

$3 {x}^{2} + 24 x - 1 \equiv 3 {\left(x + 4\right)}^{2} - 49$