Write the function f(x) = -4x2 + 24x - 42 in vertex form, and identify its vertex.

ANSWER

The vertex form is

EXPLANATION

We use completing squares

We factor -4 out of the first two terms

We add and subtract , that is coming from, .

The expression in the first parenthesis is now a perfect square.

The function is now in the form,

, where is the vertex

Hence the vertex is

the equation of a parabola in vertex form is

y = a(x - h)² + k

where (h, k ) are the coordinates of the vertex and a is a multiplier

To obtain this form use the method of completing the square

We require the coefficient of the x² term to be 1 so factor out - 4

y = - 4(x² - 6x + )

add/subtract (half the coefficient of the x-term )² to x² - 6x

y = - 4(x² + 2(- 3)x +9 - 9 +)

= - 4(x - 3)² - 4 (- 9 +)

= - 4(x - 3)² - 6 ← in vertex form

with vertex = (3, - 6)

1162-46+7=1123

52+86-9=129

1123-129= 994

994