To express the quadratic function [tex]\( f(x) = 7x^2 + 42x \)[/tex] in vertex form, we proceed through a method called "completing the square." The vertex form of a quadratic function is given by [tex]\( f(x) = a(x-h)^2 + k \)[/tex], where [tex]\((h, k)\)[/tex] is the vertex of the parabola.
Here are the steps:
1. Factor out the coefficient of [tex]\( x^2 \)[/tex] (which is 7) from the first two terms:
[tex]\[
f(x) = 7(x^2 + 6x)
\][/tex]
2. Complete the square inside the parentheses:
To complete the square, take half of the coefficient of [tex]\( x \)[/tex] (which is 6), square it, and then add and subtract this square inside the parentheses.
[tex]\[
\left(\frac{6}{2}\right)^2 = 9
\][/tex]
So, we add and subtract 9 inside the parentheses:
[tex]\[
f(x) = 7(x^2 + 6x + 9 - 9)
\][/tex]
3. Rewrite the equation by factoring the perfect square trinomial and then distribute the 7 through the remaining term:
[tex]\[
f(x) = 7[(x^2 + 6x + 9) - 9]
\][/tex]
[tex]\[
f(x) = 7[(x + 3)^2 - 9]
\][/tex]
4. Distribute the 7 to get the final form:
[tex]\[
f(x) = 7(x + 3)^2 - 7 \cdot 9
\][/tex]
[tex]\[
f(x) = 7(x + 3)^2 - 63
\][/tex]
Therefore, the function [tex]\( f(x) = 7x^2 + 42x \)[/tex] written in vertex form is:
[tex]\[
f(x) = 7(x + 3)^2 - 63
\][/tex]
The correct option is:
[tex]\[
f(x) = 7(x + 3)^2 - 63
\][/tex]
