Question

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

Answer 1

[AI-generated answer — Bot GPT 3.5]

To convert a quadratic function in standard form, \(f(x) = ax^2 + bx + c\), to vertex form, \(f(x) = a(x-h)^2 + k\), where \((h,k)\) is the vertex of the parabola, follow these steps:

1. Complete the square to rewrite the quadratic function in vertex form:
- Factor out the leading coefficient, \(a\), from the quadratic term: \(f(x) = a(x^2 + \frac{b}{a}x) + c\)
- Take half of the coefficient of \(x\), square it, and add/subtract it within the parentheses: \(f(x) = a(x^2 + \frac{b}{a}x + (\frac{b}{2a})^2 - (\frac{b}{2a})^2) + c\)
- Simplify the expression inside the parentheses: \(f(x) = a(x^2 + \frac{b}{a}x + (\frac{b}{2a})^2 - (\frac{b^2}{4a^2})) + c\)
- Factor the expression inside the parentheses: \(f(x) = a((x + \frac{b}{2a})^2 - (\frac{b^2}{4a^2})) + c\)
- Simplify: \(f(x) = a(x + \frac{b}{2a})^2 - \frac{b^2}{4a} + c\)
- Rewrite: \(f(x) = a(x + \frac{b}{2a})^2 + (\frac{-b^2 + 4ac}{4a})\)

2. The vertex form is now \(f(x) = a(x-h)^2 + k\) where
\(h = -\frac{b}{2a}\) and \(k = \frac{-b^2 + 4ac}{4a}\)

3. Replace \(h\) and \(k\) in the vertex form to get the final equation in vertex form.