analyzing the steps for conversion to vertex form\nexplain the steps necessary to convert a quadratic…

Explanation:

Step1: Start with standard form

Let the quadratic function be $y = ax^{2}+bx + c$, where $a\neq0$.

Step2: Factor out $a$ from first two terms

$y=a\left(x^{2}+\frac{b}{a}x\right)+c$.

Step3: Complete the square inside parentheses

Take half of the coefficient of $x$, square it. The coefficient of $x$ is $\frac{b}{a}$, half is $\frac{b}{2a}$, and its square is $\frac{b^{2}}{4a^{2}}$. Add and subtract $\frac{b^{2}}{4a^{2}}$ inside the parentheses: $y=a\left(x^{2}+\frac{b}{a}x+\frac{b^{2}}{4a^{2}}-\frac{b^{2}}{4a^{2}}\right)+c$.

Step4: Rewrite the trinomial as a perfect - square

$y=a\left[\left(x + \frac{b}{2a}\right)^{2}-\frac{b^{2}}{4a^{2}}\right]+c$.

Step5: Distribute $a$ and simplify

$y=a\left(x+\frac{b}{2a}\right)^{2}-\frac{b^{2}}{4a}+c$. Combine the constant terms: $y=a\left(x+\frac{b}{2a}\right)^{2}+\frac{4ac - b^{2}}{4a}$. The vertex form is $y=a(x - h)^{2}+k$, where the vertex is $(h,k)$ and $h=-\frac{b}{2a}$, $k=\frac{4ac - b^{2}}{4a}$.

Answer:

The steps are: factor out the coefficient $a$ of $x^{2}$ from the first two terms, complete the square inside the parentheses, rewrite the trinomial as a perfect - square, distribute $a$ and simplify the constant terms.