Why vertex form is useful
A quadratic can be written two ways. Standard form, y = ax² + bx + c, shows the y-intercept c at a glance. Vertex form, y = a(x − h)² + k, shows the turning point (h, k) at a glance. That makes it the form you want for graphing, for finding a maximum or minimum (the highest point of a thrown ball, the lowest cost), and for transformations.
How to complete the square
h = −b ÷ 2a k = c − b² ÷ 4a = f(h)
y = ax² + bx + c = a(x − h)² + k
y = ax² + bx + c = a(x − h)² + k
- Factor a out of the x² and x terms.
- Take half of the x coefficient inside the brackets and square it.
- Add and subtract that square inside the brackets, so nothing changes.
- The first three terms are now a perfect square. Multiply the leftover by a and combine it with c.
Going back is just expanding: a(x − h)² + k = ax² − 2ahx + (ah² + k).
Worked example
Write y = 2x² − 8x + 3 in vertex form.
- Factor out 2: y = 2(x² − 4x) + 3.
- Half of −4 is −2, and (−2)² = 4.
- y = 2(x² − 4x + 4 − 4) + 3.
- y = 2(x − 2)² − 2 × 4 + 3 = 2(x − 2)² − 5.
The vertex is (2, -5). Since a = 2 > 0 the parabola opens upward, so −5 is the minimum. The axis of symmetry is x = 2 and the y-intercept is (0, 3). Setting y = 0 gives (x − 2)² = 5/2, so x = 2 ± √10/2 ≈ 0.4189 and 3.5811.
What the numbers tell you
| Feature | From vertex form | From standard form |
|---|---|---|
| Vertex | (h, k) | (−b/2a, c − b²/4a) |
| Axis of symmetry | x = h | x = −b/2a |
| Opens | Up if a > 0 (vertex is a minimum), down if a < 0 (vertex is a maximum) | |
| y-intercept | ah² + k | c |
| x-intercepts | h ± √(−k/a), if −k/a ≥ 0 | (−b ± √(b² − 4ac)) ÷ 2a |
Limitations
- a can't be 0. Without an x² term it's a straight line, not a parabola.
- Inputs must be whole numbers, decimals or fractions so the answer can stay exact. For a coefficient like √2, use a decimal approximation.
- Complex roots aren't listed. When −k/a is negative, the page says there are no real x-intercepts.