About the Vertex Form Calculator
This vertex form calculator converts a quadratic between standard form y = ax² + bx + c and vertex form y = a(x − h)² + k, the form that shows the parabola’s turning point at a glance. Enter the coefficients a, b and c — or a, h and k if you already have vertex form — and it returns both forms, the vertex (h, k), the axis of symmetry x = h, the y-intercept and the x-intercepts (roots) if there are any.
It is useful for algebra and precalculus students practising completing the square, for teachers preparing worked solutions, and for anyone who needs the maximum or minimum of a quadratic — the peak height of a projectile, the price that maximises revenue, or the lowest point of a cable. The graph draws the parabola with its vertex in the middle.
If a is positive the parabola opens upward and the vertex is a minimum; if a is negative it opens downward and the vertex is a maximum. The value a must not be zero, because then the equation is linear rather than quadratic.
How to use the vertex form calculator
- 1Choose whether you are starting from standard form or vertex form.
- 2Enter a, b and c (or a, h and k).
- 3Read the converted equation and the vertex (h, k).
- 4Check the axis of symmetry, intercepts and whether the vertex is a max or min.
- 5Use the graph to confirm the shape and position of the parabola.
Formula and method
Vertex form comes from completing the square. Factor a out of the x terms, add and subtract (b/2a)², and the expression becomes a(x + b/2a)² + c − b²/4a. So the vertex is at h = −b/(2a) and k = c − b²/(4a), which is also f(h). Going the other way, expanding a(x − h)² + k gives b = −2ah and c = ah² + k.
The axis of symmetry is the vertical line x = h. The y-intercept is c. The x-intercepts come from the quadratic formula; the discriminant b² − 4ac tells you whether there are two real roots (positive), one repeated root at the vertex (zero) or none (negative, complex roots).
- a
- Leading coefficient — sign gives direction, size gives width
- b, c
- Linear coefficient and constant in standard form
- (h, k)
- Vertex (turning point) of the parabola
- b² − 4ac
- Discriminant — number of real roots
Worked examples
y = 2x² − 8x + 5 to vertex form
h = −(−8)/(2·2) = 2 and k = 5 − 64/8 = −3, so y = 2(x − 2)² − 3. Since a > 0 the vertex (2, −3) is a minimum. The discriminant 64 − 40 = 24 is positive, so there are two x-intercepts at 2 ± √1.5 ≈ 0.775 and 3.225.
Vertex form back to standard form
Expanding (x − 1)² − 4 gives x² − 2x + 1 − 4 = x² − 2x − 3. It factors as (x + 1)(x − 3), so the parabola crosses the x-axis at −1 and 3.
Projectile height: y = −4.9x² + 19.6x + 1.5
h = −19.6/(2 × −4.9) = 2 seconds and k = 1.5 + 19.6²/19.6 = 21.1 metres. A ball thrown upward at 19.6 m/s from 1.5 m peaks at 21.1 m after 2 s.
No real roots
h = −1 and k = 5 − 1 = 4. The vertex (−1, 4) is above the x-axis and the parabola opens upward, so it never crosses: the discriminant is 4 − 20 = −16 and the roots are −1 ± 2i.
Frequently asked questions
How do you convert standard form to vertex form?+
Compute h = −b/(2a), then k by plugging h into the equation (or k = c − b²/(4a)). Write y = a(x − h)² + k. This is the shortcut version of completing the square.
What does vertex form tell you?+
In y = a(x − h)² + k the vertex is (h, k), the axis of symmetry is x = h, and a tells you the direction and width: positive a opens up (minimum at k), negative a opens down (maximum at k), and larger |a| makes a narrower parabola.
Why is it x − h and not x + h?+
The vertex is where the squared term is zero, which happens when x = h. So y = (x − 3)² has its vertex at x = 3, while y = (x + 3)² = (x − (−3))² has its vertex at x = −3.
How do I find the x-intercepts from vertex form?+
Set y = 0 and solve: a(x − h)² = −k, so x = h ± √(−k/a). Real intercepts exist only when −k/a ≥ 0, i.e. when the vertex and the opening direction are on opposite sides of the x-axis.
Is vertex form the same as completing the square?+
Completing the square is the algebraic method; vertex form is the result. Both rewrite ax² + bx + c as a perfect square plus a constant, a(x − h)² + k.