the equation of a parabola is y = x² + 6x + 9. write the equation in vertex form. write any numbers as…

Explanation:

Step1: Recall vertex - form of parabola

The vertex - form of a parabola is $y=a(x - h)^2+k$, where $(h,k)$ is the vertex of the parabola. We will complete the square for the given quadratic function $y=x^{2}+6x + 9$.

Step2: Complete the square

For a quadratic function of the form $y=ax^{2}+bx + c$, in our case $a = 1$, $b = 6$, $c = 9$. The formula for completing the square for $ax^{2}+bx$ is $a(x+\frac{b}{2a})^{2}-\frac{b^{2}}{4a}+c$. Here, since $a = 1$, we have $y=x^{2}+6x + 9=(x^{2}+6x)+9$. We know that $(x + m)^2=x^{2}+2mx+m^{2}$, for $x^{2}+6x$, if $2m = 6$, then $m = 3$ and $x^{2}+6x=(x + 3)^{2}-9$. So $y=(x + 3)^{2}-9+9$.

Step3: Simplify the equation

$y=(x + 3)^{2}-9 + 9=(x+3)^{2}+0$.

Answer:

$y=(x + 3)^{2}$