Question

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

Answer 1

ANSWER

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

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, $(\frac{b}{2a})^2$.

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

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

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$(3,-6)$

Answer 2

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 + $\frac{42}{4}$)

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

y = - 4(x² + 2(- 3)x +9 - 9 +$\frac{21}{2}$)

= - 4(x - 3)² - 4 (- 9 +$\frac{21}{2}$)

= - 4(x - 3)² - 6 ← in vertex form

with vertex = (3, - 6)