Final answer:
Rewriting a quadratic equation in vertex form involves completing the square, which leads to y = a(x-h)^2 + k, where (h, k) is the vertex of the parabola. Converting it back to standard form simply requires expanding and combining like terms.
Step-by-step explanation:
When rewriting a quadratic equation in vertex form, you are essentially completing the square to express the function as y = a(x - h)^2 + k, where (h, k) is the vertex of the parabola. To start, let's take an equation y = ax^2 + bx + c and rewrite it:
- First, factor out the coefficient of x^2 from the first two terms if it's not 1.
- Next, take half the coefficient of x (which is b/2a), square it, add and subtract this squared term inside the parentheses.
- Rewrite the equation by combining the constant terms outside the parentheses.
- This results in the vertex form: y = a(x-h)^2 + k where h = -b/(2a) and k is the constant term that results from the completing the square process.
To rewrite the function in standard form, you then expand the vertex form back into the general form, simply performing the multiplication and combining like terms.
If we use a specific quadratic equation, for example, y = 1.00x^2 + 10.0x - 200, we can find its vertex form following these steps:
- Since the coefficient of x^2 is 1, we do not need to factor anything out.
- The coefficient of x is 10.0, so we take half of it (5.0), square it (25.0), add and subtract it inside the equation.
- Add and subtract 25.0 to get y = x^2 + 10.0x + 25.0 - 25.0 - 200.
- Group the perfect square trinomial: y = (x^2 + 10.0x + 25.0) - 225.
- Factor the trinomial: y = (x + 5)^2 - 225, which is the vertex form with the vertex at (-5, -225).
To convert it back to standard form, expand (x + 5)^2 to x^2 + 10.0x + 25.0, then combine like terms with -225 to get back the original standard form y = x^2 + 10.0x - 200.