Certainly! Let's solve the given quadratic equation step by step to find the vertex of the parabola defined by [tex]\( y = -x^2 - 4x + 4 \)[/tex].
The standard form of a quadratic equation is [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]\( y = -x^2 - 4x + 4 \)[/tex], we can see that:
- [tex]\( a = -1 \)[/tex],
- [tex]\( b = -4 \)[/tex],
- [tex]\( c = 4 \)[/tex].
To find the vertex of the parabola, we use the vertex formula for the x-coordinate:
[tex]\[ x = -\frac{b}{2a} \][/tex]
Substitute [tex]\( a = -1 \)[/tex] and [tex]\( b = -4 \)[/tex] into the formula:
[tex]\[ x = -\frac{-4}{2(-1)} \][/tex]
[tex]\[ x = \frac{4}{-2} \][/tex]
[tex]\[ x = -2 \][/tex]
Now that we have the x-coordinate of the vertex, we need to find the y-coordinate. We substitute [tex]\( x = -2 \)[/tex] back into the original equation:
[tex]\[ y = -(-2)^2 - 4(-2) + 4 \][/tex]
[tex]\[ y = -4 + 8 + 4 \][/tex]
[tex]\[ y = 8 - 4 \][/tex]
[tex]\[ y = 4 \][/tex]
Therefore, the vertex of the parabola [tex]\( y = -x^2 - 4x + 4 \)[/tex] is at:
[tex]\[ (-2, 8) \][/tex]
To summarize:
- The x-coordinate of the vertex is [tex]\( -2 \)[/tex],
- The y-coordinate of the vertex is [tex]\( 8 \)[/tex].
So, the vertex is at [tex]\( (-2, 8) \)[/tex].
