Let's convert the quadratic expression [tex]\( y = x^2 + 2x - 1 \)[/tex] to its vertex form step-by-step.
1. Start with the given quadratic equation:
[tex]\[
y = x^2 + 2x - 1
\][/tex]
2. Form a perfect-square trinomial:
We need to complete the square inside the expression. To do this, we look at the coefficient of the [tex]\(x\)[/tex]-term which is [tex]\(2\)[/tex].
[tex]\[
x^2 + 2x
\][/tex]
To complete the square, add and then subtract the square of half the coefficient of [tex]\(x\)[/tex]:
[tex]\[
x^2 + 2x + 1 - 1 = (x+1)^2 - 1
\][/tex]
3. Incorporate the constant term:
Now, bring down the [tex]\(-1\)[/tex] originally in the equation and adjust for the [tex]\(-1\)[/tex] we just subtracted:
[tex]\[
y = (x + 1)^2 - 2
\][/tex]
4. Write the final vertex form:
The quadratic expression in vertex form is:
[tex]\[
y = (x + 1)^2 - 2
\][/tex]
In this form, [tex]\((x + 1)^2 - 2\)[/tex], the vertex of the parabola is at the point [tex]\((-1, -2)\)[/tex]. The vertex form makes it clear how the parabola opens and where its vertex is located.
