To identify the vertex of the quadratic equation [tex]\( y = x^2 + 4x + 3 \)[/tex], let's follow these steps:
1. Understand the standard form of a quadratic equation:
A quadratic equation is typically written in the form [tex]\( y = ax^2 + bx + c \)[/tex], where:
- [tex]\( a \)[/tex] is the coefficient of [tex]\( x^2 \)[/tex]
- [tex]\( b \)[/tex] is the coefficient of [tex]\( x \)[/tex]
- [tex]\( c \)[/tex] is the constant term
For the given equation [tex]\( y = x^2 + 4x + 3 \)[/tex]:
- [tex]\( a = 1 \)[/tex]
- [tex]\( b = 4 \)[/tex]
- [tex]\( c = 3 \)[/tex]
2. Identify the x-coordinate of the vertex:
The x-coordinate of the vertex can be found using the formula [tex]\( x = -\frac{b}{2a} \)[/tex].
Substitute the values of [tex]\( a \)[/tex] and [tex]\( b \)[/tex] into the formula:
[tex]\[
x = -\frac{4}{2 \cdot 1} = -\frac{4}{2} = -2
\][/tex]
3. Identify the y-coordinate of the vertex:
Once we have the x-coordinate, we substitute [tex]\( x = -2 \)[/tex] back into the original equation to find the y-coordinate.
Substitute [tex]\( x = -2 \)[/tex] into [tex]\( y = x^2 + 4x + 3 \)[/tex]:
[tex]\[
y = (-2)^2 + 4(-2) + 3
\][/tex]
Calculate each term:
[tex]\[
y = 4 - 8 + 3 = -1
\][/tex]
4. Determine the vertex:
With the x-coordinate as [tex]\(-2\)[/tex] and the y-coordinate as [tex]\(-1\)[/tex], the vertex of the quadratic equation [tex]\( y = x^2 + 4x + 3 \)[/tex] is:
[tex]\[
(-2, -1)
\][/tex]
5. Identify the correct option:
Based on the options provided:
- [tex]\( \boxed{(-2, -1)} \)[/tex] is the correct answer.
So, the vertex of the equation [tex]\( y = x^2 + 4x + 3 \)[/tex] is [tex]\( (-2, -1) \)[/tex]. Therefore, the correct option is B.
