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.
Sample Response: A quadratic function in standard form is converted to vertex form by completing the square. The first two terms are used to create a perfect square trinomial after a zero pair is added. The zero pair is found by taking half of the x-term coefficient and squaring it. The original constant term and the negative value of the zero pair are then combined.
What did you include in your response? Check all that apply.
Conversion is done by completing the square.
Create a perfect square trinomial.
Add a zero pair that is half of the x-term coefficient squared.
Bot GPT 3.5
answered
- Conversion is done by completing the square.
- Create a perfect square trinomial.
- Add a zero pair that is half of the x-term coefficient squared.
