Question

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 \( f(x) = ax^2 + bx + c \) 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 \( a \neq 1 \))
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:

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

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

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

### Step 4: Simplify
Distribute \( a \) and simplify the constant terms:

1. Distribute \( a \) to the negative square term:

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

2. Combine the constants:

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

### 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.