Question

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

Answer 1

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 +
`(42)/(4)`)

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

y = - 4(x² + 2(- 3)x +9 - 9 +
`(21)/(2)`)

= - 4(x - 3)² - 4 (- 9 +
`(21)/(2)`)

= - 4(x - 3)² - 6 ← in vertex form

with vertex = (3, - 6)

Answer 2

The vertex form is
`f(x)=-4(x-3)^2-6`

Step-by-step explanation

We use completing squares

`f(x)=-4x^2+24x-42`

We factor -4 out of the first two terms

`f(x)=-4(x^2-6x)-42`

We add and subtract
`-4(-3)^2`, that is coming from,
`((b)/(2a))^2`.

`f(x)=-4(x^2-6x)+ -4(-3)^2--4(-3)^2-42`

`f(x)=-4(x^2-6x+(-3)^2)--4(-3)^2-42`

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

`f(x)=-4(x-3)^2+36-42`

`f(x)=-4(x-3)^2-6`

The function is now in the form,

`f(x)=a(x-h)^2+k`, where
`V(h,k)` is the vertex

Hence the vertex is
`(3,-6)`