Vertex Form Calculator
Convert a quadratic between standard form y = ax² + bx + c and vertex form y = a(x - h)² + k, or enter the vertex directly. The calculator finds the vertex, x-intercepts, y-intercept, axis of symmetry, focus, and discriminant with a full show-your-work breakdown.
Formula
Worked example
For y = x^2 - 6x + 5 (a = 1, b = -6, c = 5): h = -(-6) / (2 x 1) = 3, k = 5 - 36/4 = -4. Vertex form: y = (x - 3)^2 - 4. Discriminant D = 36 - 20 = 16 > 0, so x-intercepts are x = (6 +/- 4) / 2 = 5 and x = 1. Y-intercept = 5. Focus = (3, -3.75).
From standard form to vertex form
A quadratic written as y = ax^2 + bx + c is in standard form, which is convenient for reading the y-intercept c but hides the turning point of the parabola. Vertex form, y = a(x - h)^2 + k, makes that turning point explicit: the vertex sits at coordinates (h, k). Converting between them involves completing the square, but the shortcut is to compute the axis of symmetry h = -b / (2a) and then evaluate k = c - b^2 / (4a), which equals the function value at x = h. The leading coefficient a is identical in both forms, controlling width and direction without changing during the conversion.
From vertex form back to standard form
Going the other direction, given y = a(x - h)^2 + k, expand the squared binomial: (x - h)^2 = x^2 - 2hx + h^2. Multiply through by a to get ax^2 - 2ahx + ah^2, then add k to recover b = -2ah and c = ah^2 + k. This is the reverse-solve mode on this calculator: enter a, h, and k in vertex mode to get the standard-form coefficients immediately.
Roots, y-intercept, and discriminant
The y-intercept is simply the function evaluated at x = 0, which equals ah^2 + k (the same as the constant c in standard form). The x-intercepts, or roots, exist only when the discriminant D = b^2 - 4ac is non-negative. When D > 0 there are two distinct roots found by the quadratic formula; when D = 0 the parabola is tangent to the x-axis at exactly one point; when D < 0 the roots are complex and the parabola never crosses the x-axis. The calculator displays all three scenarios and gives the exact root values when they exist.
Axis of symmetry and focus
The axis of symmetry is the vertical line x = h, the mirror line that makes both sides of the parabola identical. In optics and engineering, the focus of a parabola is a special point where incoming parallel rays reflect to a single point. For y = a(x - h)^2 + k, the focal length is p = 1 / (4a), and the focus sits at (h, k + p). A longer focal length (small |a|) gives a wide, shallow parabola; a shorter focal length (large |a|) gives a narrow, steep one.
Standard form versus vertex form: worked examples
| Standard form | Vertex (h, k) | Vertex form | Roots |
|---|---|---|---|
| y = x^2 - 6x + 5 | (3, -4) | y = (x - 3)^2 - 4 | x = 1, x = 5 |
| y = 2x^2 + 8x + 3 | (-2, -5) | y = 2(x + 2)^2 - 5 | x ~ -0.42, x ~ -3.58 |
| y = -x^2 + 4x - 1 | (2, 3) | y = -(x - 2)^2 + 3 | x ~ 0.27, x ~ 3.73 |
| y = x^2 + 4 | (0, 4) | y = x^2 + 4 | No real roots |
| y = x^2 - 4x + 4 | (2, 0) | y = (x - 2)^2 | x = 2 (repeated) |
The same parabola written two ways. Roots exist when D >= 0.
Frequently asked questions
How do I find h and k from a, b and c?
Compute h = -b / (2a) for the x-coordinate of the vertex, then substitute back to find k = c - b^2 / (4a). Alternatively, evaluate the original function at x = h to get k. The pair (h, k) is the vertex.
How do I convert vertex form back to standard form?
Expand (x - h)^2 = x^2 - 2hx + h^2, multiply by a, and add k. This gives b = -2ah and c = ah^2 + k. Use the Vertex form input mode on this calculator to do it automatically.
Does the value of a change when I convert between forms?
No. The leading coefficient a is the same in y = ax^2 + bx + c and y = a(x - h)^2 + k. It controls the width and opening direction of the parabola, which the conversion never changes.
How do I find the x-intercepts (roots) from vertex form?
Set y = 0 and solve: a(x - h)^2 + k = 0 gives (x - h)^2 = -k/a, so x = h +/- sqrt(-k/a). Roots are real only when -k/a >= 0, which happens when a and k have opposite signs (or k = 0). The discriminant D = b^2 - 4ac tells you the same thing.
What is the axis of symmetry?
The axis of symmetry is the vertical line x = h. Every parabola is perfectly symmetric about this line. It passes through the vertex and divides the curve into two mirror-image halves.
What is the focus of a parabola?
The focus is a special point inside the parabola used in optics, satellite dishes, and physics. For y = a(x - h)^2 + k, the focal length is p = 1 / (4a) and the focus is located at (h, k + p). A parabolic reflector focuses all incoming parallel rays to the focus.
What happens if a is zero?
If a = 0 the expression is linear, not quadratic. It has no vertex and no parabola. Vertex form requires a non-zero leading coefficient, so this calculator returns no result when a = 0.