To transform the function [tex]\( f(x) = 6x^2 + 5 - 42x \)[/tex] into vertex form, we first need to rewrite it in a useful intermediate form. Here's how we can do it step-by-step:
1. Write the function in standard form:
The given function is already in standard form as [tex]\( f(x) = 6x^2 + 5 - 42x \)[/tex]. To better see the terms, we can reorganize it:
[tex]\[
f(x) = 6x^2 - 42x + 5
\][/tex]
2. Factor out the leading coefficient from the quadratic and linear terms:
The standard form of a quadratic function is [tex]\( ax^2 + bx + c \)[/tex], and here [tex]\( a = 6 \)[/tex], [tex]\( b = -42 \)[/tex], and [tex]\( c = 5 \)[/tex].
To complete the square, we start by factoring out the coefficient [tex]\( 6 \)[/tex] from the quadratic term and the linear term:
[tex]\[
f(x) = 6(x^2 - 7x) + 5
\][/tex]
Thus, the first step in writing [tex]\( f(x) = 6x^2 + 5 - 42x \)[/tex] in vertex form is to factor 6 out of the first two terms of the quadratic expression.
