To rewrite the quadratic function [tex]\( f(x) = 7x^2 + 42x \)[/tex] in vertex form, we need to complete the square. The vertex form of a quadratic function is [tex]\( f(x) = a(x - h)^2 + k \)[/tex], where [tex]\((h, k)\)[/tex] is the vertex of the parabola.
Let's follow these steps carefully:
1. Factor out the coefficient of [tex]\( x^2 \)[/tex] from the first two terms:
[tex]\[ f(x) = 7(x^2 + 6x) \][/tex]
2. Complete the square inside the parenthesis:
- Take the coefficient of [tex]\( x \)[/tex], which is 6, and halve it to get 3.
- Square 3 to get [tex]\( 3^2 = 9 \)[/tex].
So, we can rewrite the trinomial by adding and subtracting [tex]\( 9 \)[/tex] inside the parenthesis:
[tex]\[ f(x) = 7(x^2 + 6x + 9 - 9) \][/tex]
[tex]\[ f(x) = 7((x^2 + 6x + 9) - 9) \][/tex]
[tex]\[ f(x) = 7((x + 3)^2 - 9) \][/tex]
3. Distribute the 7 across the terms inside the parenthesis:
[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]
Hence, the correct answer is:
[tex]\[ f(x) = 7(x + 3)^2 - 63 \][/tex]
So, the correct choice is:
[tex]\[ \boxed{f(x)=7(x+3)^2-63} \][/tex]
