Parabola Vertex, Focus & Roots Calculator

Parabola Guide

What should I know about Forms of a Parabola Equation?

Standard form: y = ax² + bx + c. Most common in school maths. a determines opening direction (a > 0: up, a < 0: down) and width (larger |a|: narrower). Vertex form: y = a(x − h)² + k. Directly shows vertex at (h, k). Useful for graphing and transformations. Factored form: y = a(x − r₁)(x − r₂). Shows the roots r₁ and r₂ directly. Only works when roots are real. Converting standard to vertex form: complete the square. y = a(x² + (b/a)x) + c = a(x + b/(2a))² + c − b²/(4a). So vertex: h = −b/(2a), and k is found by substituting this h value back into the original equation — the same result the completing-the-square method arrives at directly.

What should I know about Vertex, Axis, and Roots?

Vertex: the maximum or minimum point. x-coordinate: h = −b/(2a). y-coordinate: substitute h back into equation. Axis of symmetry: vertical line through vertex. Equation: x = −b/(2a). Roots (x-intercepts): solve y = 0. Use quadratic formula: x = (−b ± √(b² − 4ac)) / (2a). Sum of roots: −b/a. Product of roots: c/a. Useful for verifying answers. y-intercept: substitute x = 0. y = c. The y-intercept is always the constant term in standard form. Example: y = x² − 4x + 3. Vertex: h = 4/2 = 2, k = 3 − 4 = −1, giving a vertex at (2, −1) for this particular parabola.

What should I know about Focus and Directrix?

A parabola is defined geometrically: every point is equidistant from focus and directrix. Focus: a fixed point inside the parabola curve. Directrix: a line outside, perpendicular to the axis of symmetry. For y = ax² + bx + c with vertex (h, k): focal length p = 1/(4a). Focus: (h, k + p). Directrix: y = k − p. Larger |a|: focus closer to vertex (narrower parabola). Smaller |a|: focus further from vertex (wider). Applications of focus property: parabolic reflectors, such as satellite dishes, headlamps, and microphone dishes, all exploit the property that signals or light entering parallel to the axis reflect through the single focus point.

What should I know about Real-World Parabolas?

Projectile motion: any thrown object follows a parabolic path (ignoring air resistance). h(t) = h₀ + v₀t − ½gt². Range: maximum at 45° launch angle (in vacuum). With air resistance: paths are slightly distorted but still close to parabolic for moderate speeds. Architecture: arches, bridges, and structural supports often use parabolic curves for efficient load distribution. Hanging cables: form a catenary (very close to but not exactly a parabola). Parabolic bridges, such as concrete viaducts, use the shape specifically for its load-distributing structural properties, spreading weight evenly along the curve.

What should I know about Completing the Square to Find the Vertex?

Any quadratic in standard form y = ax² + bx + c can be rewritten in vertex form y = a(x - h)² + k by completing the square, which directly reveals the vertex (h, k). The steps: factor a out of the x² and x terms, add and subtract (b/2a)² inside the bracket to make a perfect square, then simplify. For y = 2x² + 8x + 3, factor out 2 to get 2(x² + 4x) + 3, complete the square inside (half of 4 is 2, squared is 4) to get 2(x² + 4x + 4 - 4) + 3 = 2(x + 2)² - 8 + 3 = 2(x + 2)² - 5, giving vertex (-2, -5). This matches the shortcut formula h = -b/2a, k = c - b²/4a, but completing the square shows exactly where that formula comes from.

What should I know about Maximum and Minimum Values in Optimisation Problems?

Because a parabola's vertex is its single highest or lowest point, finding the vertex is the standard way to solve "maximize or minimize" word problems that reduce to a quadratic. If a > 0, the parabola opens upward and the vertex gives the minimum value of y; if a < 0, it opens downward and the vertex gives the maximum. A classic example: a farmer with 100m of fencing wants to maximize a rectangular field's area against a wall (so only three sides need fencing). Writing area as a function of one side length produces a downward-opening parabola, and the vertex's x-coordinate gives the optimal side length while the y-coordinate gives the maximum area achievable. This same logic underlies profit-maximization and projectile-height problems in business and physics.

What should I know about the Discriminant and Real Roots?

For a quadratic ax² + bx + c, the discriminant Δ = b² − 4ac determines not just how many real solutions the equation ax² + bx + c = 0 has, but also exactly how the corresponding parabola sits relative to the x-axis. When Δ > 0, the quadratic formula produces two distinct real roots, meaning the parabola crosses the x-axis at two separate points. When Δ = 0, the two roots coincide into a single repeated root, which means the parabola just touches the x-axis at exactly one point — and that point is precisely the vertex of the parabola, since a parabola with only one x-intercept must be tangent to the axis there. When Δ < 0, the quadratic formula requires taking the square root of a negative number, giving two complex conjugate roots with no real solutions, which means the parabola never touches the x-axis at all: it lies entirely above the axis (opening upward with vertex above the x-axis, when a > 0) or entirely below it (opening downward with vertex below the x-axis, when a < 0). This connection between the discriminant's sign and the parabola's position relative to the x-axis makes it possible to determine how many x-intercepts a parabola has without solving the quadratic equation at all.

Worked Example: Completing the Square for y = 2x² - 8x + 5

To find the vertex of y = 2x² − 8x + 5 by completing the square, first factor the coefficient of x² out of the first two terms: y = 2(x² − 4x) + 5. Take half of the coefficient of x inside the brackets (−4 ÷ 2 = −2), square it (−2² = 4), add and subtract it inside: y = 2(x² − 4x + 4 − 4) + 5 = 2[(x−2)² − 4] + 5. Distribute the 2: y = 2(x−2)² − 8 + 5 = 2(x−2)² − 3. This is vertex form, y = a(x−h)² + k, so the vertex is at (h, k) = (2, −3). Since a = 2 is positive, the parabola opens upward and (2, −3) is a minimum point. Checking by substituting x = 2 into the original equation: y = 2(4) − 8(2) + 5 = 8 − 16 + 5 = −3, which matches.

What the Discriminant Tells You About the Roots

Before solving a quadratic with the formula, the discriminant (the part under the square root, b² − 4ac) reveals how many real solutions exist and what kind they are, without doing the full calculation.

For y = 2x² − 8x + 5 from the worked example above, the discriminant is (−8)² − 4(2)(5) = 64 − 40 = 24, which is positive, so the parabola does cross the x-axis at two points, consistent with having a minimum value of −3 below the axis.

Vertex Form vs Standard Form: When to Use Each

Standard form, y = ax² + bx + c, is usually how a parabola is first given and is convenient for reading off the y-intercept directly (it's just c). Vertex form, y = a(x−h)² + k, is far more convenient once the vertex is needed, since (h, k) can be read straight off without further calculation. Converting from standard to vertex form (via completing the square or the vertex formula h = -b/2a) is exactly the step a vertex calculator automates, trading a form that's easy to write down for one that's easy to interpret.

What the Discriminant Reveals About the Roots

For a quadratic ax² + bx + c = 0, the discriminant (b² − 4ac) tells you how many real roots exist before you even solve the equation — directly connected to where the parabola's vertex sits relative to the x-axis.

If the vertex y-value calculated by this tool is positive and the parabola opens upward, the discriminant must be negative — the two facts are just different views of the same geometry, so checking one is a quick way to sanity-check the other.