Certainly! Let's find the vertex of the parabola given by the quadratic equation \( y = x^2 + 2x - 3 \).
For a quadratic equation of the form \( y = ax^2 + bx + c \), the vertex can be found using the vertex formula. The coordinates of the vertex (h, k) are given by:
[tex]\[ x = -\frac{b}{2a} \][/tex]
[tex]\[ y = a(x)^2 + b(x) + c \][/tex]
In this equation, the coefficients are:
[tex]\[ a = 1 \][/tex]
[tex]\[ b = 2 \][/tex]
[tex]\[ c = -3 \][/tex]
1. Calculate the x-coordinate of the vertex (h):
[tex]\[ h = -\frac{b}{2a} \][/tex]
[tex]\[ h = -\frac{2}{2(1)} \][/tex]
[tex]\[ h = -\frac{2}{2} \][/tex]
[tex]\[ h = -1.0 \][/tex]
So, the x-coordinate of the vertex is \( x = -1.0 \).
2. Now, substitute \( x = -1.0 \) back into the original equation to find the y-coordinate (k):
[tex]\[ k = a(h)^2 + b(h) + c \][/tex]
[tex]\[ k = 1(-1.0)^2 + 2(-1.0) - 3 \][/tex]
[tex]\[ k = 1(1) + 2(-1) - 3 \][/tex]
[tex]\[ k = 1 - 2 - 3 \][/tex]
[tex]\[ k = -4.0 \][/tex]
Therefore, the y-coordinate of the vertex is \( y = -4.0 \).
So, the vertex of the parabola described by the equation [tex]\( y = x^2 + 2x - 3 \)[/tex] is [tex]\((-1.0, -4.0)\)[/tex].
