To convert the quadratic function [tex]\( f(x) = 7x^2 + 42x \)[/tex] into its vertex form, follow these steps:
1. Identify the coefficients:
Here, the quadratic equation is [tex]\( f(x) = ax^2 + bx + c \)[/tex]. For the given function, we have:
- [tex]\( a = 7 \)[/tex]
- [tex]\( b = 42 \)[/tex]
- [tex]\( c = 0 \)[/tex] (since the function does not have a constant term)
2. Find the vertex [tex]\( (h, k) \)[/tex]:
The vertex [tex]\( (h, k) \)[/tex] of a quadratic function [tex]\( ax^2 + bx + c \)[/tex] can be found using the formula for the [tex]\( x \)[/tex]-coordinate of the vertex [tex]\( h = -\frac{b}{2a} \)[/tex].
Substituting the given values:
[tex]\[ h = -\frac{b}{2a} = -\frac{42}{2 \cdot 7} = -\frac{42}{14} = -3 \][/tex]
3. Calculate the [tex]\( y \)[/tex]-coordinate [tex]\( k \)[/tex]:
Substitute [tex]\( h = -3 \)[/tex] back into the function to find [tex]\( k \)[/tex]:
[tex]\[ k = f(-3) = 7(-3)^2 + 42(-3) \][/tex]
[tex]\[ k = 7 \cdot 9 - 42 \cdot 3 \][/tex]
[tex]\[ k = 63 - 126 \][/tex]
[tex]\[ k = -63 \][/tex]
4. Write the vertex form:
Now, the vertex form of a quadratic function [tex]\( f(x) = a(x - h)^2 + k \)[/tex] is:
[tex]\[ f(x) = 7(x - (-3))^2 + (-63) \][/tex]
[tex]\[ f(x) = 7(x + 3)^2 - 63 \][/tex]
Thus, the given quadratic function [tex]\( f(x) = 7x^2 + 42x \)[/tex] in vertex form is:
[tex]\[ f(x) = 7(x + 3)^2 - 63 \][/tex]
This corresponds to the option:
[tex]\[ f(x) = 7(x + 3)^2 - 63 \][/tex]
So, the correct answer is:
[tex]\[ \boxed{4} \][/tex]
