The Most Suitable Form of a Quadratic
A quadratic can be written three ways — vertex, factored, or standard form — and each one puts a different feature of the parabola on display without any extra work.
A quadratic is one function, but its formula can be rearranged to highlight whatever feature you need. Vertex form, a(x - h)2 + k, hands you the vertex (h, k) directly, because the squared term is zero exactly when x = h and only adds height otherwise. Factored form, a(x - p)(x - q), hands you the zeros p and q, since the product is zero the moment either factor is zero. Standard form, ax2 + bx + c, hands you the y-intercept c, because plugging in x = 0 kills the other two terms instantly. None of the three is more correct than the others. They're the same parabola wearing different outfits, and the question tells you which outfit to ask for.
The catch is that standard form only displays the y-intercept. The vertex is hiding in there, but you can't just read it off the coefficients b and c. To expose it, you complete the square: rewrite x2 - bx + c as (x - b/2)2 + c - b2/4. That's exactly why vertex form exists — it's the output of completing the square, pre-packaged so you never have to redo that algebra. The vertex sits at x = b/2 (half the coefficient of x, with the sign flipped from how it appears in standard form), not at (b, c).
This connects to the rest of advanced math because switching forms is often the whole problem in disguise: a question about a minimum or maximum value is a vertex-form question, a question about where a graph crosses the x-axis is a factored-form question, and recognizing which form a problem is secretly asking for saves you from a much longer algebraic path.
Key terms
- Vertex form
- a(x - h)2 + k, which displays the vertex (h, k) directly.
- Factored form
- a(x - p)(x - q), which displays the zeros p and q directly.
- Standard form
- ax2 + bx + c, which displays the y-intercept c directly.
- Completing the square
- the algebra that converts standard form into vertex form by forcing a perfect square trinomial.
- Vertex
- the point where a parabola reaches its minimum or maximum value.
Watch it explained
- How it worksHow does vertex form hand over the vertex, factored form the zeros, and standard form the y-intercept?
- Worked exampleFind the minimum value of g(x) = x2 + 55 by naming the form that puts it on display.
- Worked exampleShow me f(x) = x2 - 10x + 21 rewritten first to display its vertex, then to display its zeros.
- Common mistakeWhy doesn't standard form reveal the vertex even though all three numbers are sitting right there?
- Ties togetherHow does completing the square explain why vertex form exists in the first place?
Check yourself
End of topic.