Explain the steps necessary to convert a quadratic function in standard form to vertex form.

Answer 1

To convert a quadratic function from standard form $f(x)=a{x}^2+bx+c$ to vertex form $f(x)=a(x-h{)}^2+k$, you can follow these steps:

Step 1: Identify $a$, $b$, and $c$

Start by identifying the coefficients $a$, $b$, and $c$ in your standard form equation.

Step 2: Factor out $a$ (if $a\ne 1$)

If $a$ is not equal to 1, factor it out of the first two terms. This gives you:

$$f(x)=a(x^2+\frac{b}{a}x)+c$$

Step 3: Complete the square

To complete the square inside the parentheses:

1. Take the coefficient of $x$ (which is $\frac{b}{a}$), divide it by 2, and square it. This value will be added and subtracted inside the parentheses.

   $${\left(\frac{b}{2a}\right)}^2=\frac{b^2}{4a^2}$$

2. Rewrite the function by adding and subtracting this square inside the parentheses:

   $$f(x)=a(x^2+\frac{b}{a}x+\frac{b^2}{4a^2}-\frac{b^2}{4a^2})+c$$

3. This allows you to factor the perfect square trinomial:

   $$f(x)=a({(x+\frac{b}{2a})}^2-\frac{b^2}{4a^2})+c$$

Step 4: Simplify

Distribute $a$ and simplify the constant terms:

1. Distribute $a$ to the negative square term:

$$f(x)=a{(x+\frac{b}{2a})}^2-a\frac{b^2}{4a^2}+c$$

2. Combine the constants:

$$f(x)=a{(x+\frac{b}{2a})}^2+(c-\frac{b^2}{4a})$$

Step 5: Identify $h$ and $k$

Now, from the vertex form $f(x)=a(x-h{)}^2+k$:

• $h=-\frac{b}{2a}$
• $k=c-\frac{b^2}{4a}$

Conclusion

The function is now in vertex form $f(x)=a(x-h{)}^2+k$, where you have determined the vertex $(h,k)$.

Putting it all together, the process involves identifying coefficients, factoring out $a$, completing the square, and finally simplifying to find the vertex form.