To find the vertex of the quadratic function [tex]\( f(x) = x^2 + 4x \)[/tex], we can follow these steps:
1. Identify the coefficients: In the quadratic equation [tex]\( ax^2 + bx + c \)[/tex], we need to identify the coefficients [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex]. For the function [tex]\( f(x) = x^2 + 4x \)[/tex]:
- [tex]\( a = 1 \)[/tex]
- [tex]\( b = 4 \)[/tex]
- [tex]\( c = 0 \)[/tex] (since there is no constant term)
2. Find the x-coordinate of the vertex: The x-coordinate of the vertex of a parabola given by the equation [tex]\( ax^2 + bx + c \)[/tex] can be found using the formula:
[tex]\[
h = -\frac{b}{2a}
\][/tex]
Substituting the identified values of [tex]\( a \)[/tex] and [tex]\( b \)[/tex]:
[tex]\[
h = -\frac{4}{2 \cdot 1} = -\frac{4}{2} = -2
\][/tex]
3. Find the y-coordinate of the vertex: To find the y-coordinate [tex]\( k \)[/tex] of the vertex, substitute the x-coordinate [tex]\( h \)[/tex] back into the original function:
[tex]\[
k = f(h) = f(-2)
\][/tex]
Substituting [tex]\( -2 \)[/tex] into the function [tex]\( f(x) = x^2 + 4x \)[/tex]:
[tex]\[
f(-2) = (-2)^2 + 4(-2) = 4 - 8 = -4
\][/tex]
4. Determine the vertex: The vertex of the parabola [tex]\( f(x) = x^2 + 4x \)[/tex] is the point [tex]\( (h, k) \)[/tex]:
[tex]\[
\text{Vertex} = (-2, -4)
\][/tex]
Therefore, the correct vertex from the given options is:
[tex]\[
(-2, -4)
\][/tex]
