Vertex Function Calculator
Use standard coefficients or vertex-form values to study a parabola often used for projectile-style physics curves.
a creates a peak, while a positive a creates a valley.
Plotly Graph
The curve highlights how the vertex controls the turning point, symmetry, and overall shape of the parabola.
Example Data Table
| Scenario | Equation | Vertex | Axis | Real Roots | Direction |
|---|---|---|---|---|---|
| Projectile arc sample | y = -0.18x² + 2.40x + 1.50 | (6.67, 9.50) | x = 6.67 | -0.58, 13.91 | Downward |
| Reflector profile | y = 0.25x² - 2.00x + 6.00 | (4.00, 2.00) | x = 4.00 | No real roots | Upward |
| Symmetric launch path | y = -0.50x² + 4.00x | (4.00, 8.00) | x = 4.00 | 0.00, 8.00 | Downward |
Formula Used
Standard form: y = ax² + bx + c
Vertex x-coordinate: xv = -b / 2a
Vertex y-coordinate: yv = f(xv)
Vertex form: y = a(x - h)² + k, where the vertex is (h, k)
Discriminant: Δ = b² - 4ac
Roots: x = (-b ± √Δ) / 2a
Focus shift: p = 1 / 4a, so focus = (h, k + p) and directrix = y = k - p
In physics, the vertex often represents maximum height, minimum potential, or a turning position in a parabola-based model.
How to Use This Calculator
- Choose whether you want to enter the curve in standard form or vertex form.
- Type the coefficient values and optional axis labels.
- Use a unit label such as m, ft, s, or cm for cleaner reporting.
- Press Calculate Vertex Function to solve the curve.
- Review the result card above the form for the solved values.
- Inspect the Plotly graph to see the turning point and symmetry visually.
- Download the solved summary as CSV or PDF when needed.
FAQs
1. What does the vertex represent in physics?
The vertex is the turning point of the parabola. In motion problems, it often gives the maximum height of a projectile or the lowest point of a modeled energy curve.
2. Why must coefficient a be non-zero?
If a equals zero, the equation becomes linear instead of quadratic. A true vertex only exists for a parabola, so this calculator requires a non-zero value.
3. What does a negative a mean?
A negative a means the parabola opens downward. That creates a maximum point at the vertex, which is common in projectile motion graphs.
4. What does a positive a mean?
A positive a means the parabola opens upward. The vertex becomes a minimum point, useful for reflector shapes, energy wells, or position curves.
5. What is the discriminant used for?
The discriminant shows whether real roots exist. A positive value gives two real roots, zero gives one repeated root, and a negative value gives no real x-axis intersections.
6. When should I use vertex form?
Use vertex form when you already know the turning point and curve width. It is especially convenient for modeling peak height and horizontal location directly.
7. Can this calculator help with trajectory graphs?
Yes. If your trajectory can be modeled by a quadratic equation, the calculator reveals the peak, symmetry axis, intercepts, and curve shape on the graph.
8. What do the CSV and PDF exports include?
They include the solved equation details, vertex data, focus, directrix, and sampled graph points. This makes it easier to save, share, or compare curve results.