Calculator
Formula used
Standard form is y = ax^2 + bx + c. Vertex form is y = a(x - h)^2 + k, where:
- h = -b / (2a)
- k = c - b^2 / (4a)
For vertex to standard: b = -2ah and c = ah^2 + k.
How to use this calculator
- Select a conversion mode.
- Enter the required parameters (a must not be zero).
- Press Submit to show results above the form.
- Download CSV or PDF for reports or assignments.
- Reset to start over anytime.
Example data table
| Input form | Parameters | Vertex (h, k) | Vertex form |
|---|---|---|---|
| Standard | a=2, b=-8, c=5 | (2, -3) | 2(x - 2)^2 - 3 |
| Standard | a=1, b=6, c=9 | (-3, 0) | (x + 3)^2 + 0 |
| Vertex | a=-1, h=1, k=4 | (1, 4) | -(x - 1)^2 + 4 |
Article
Vertex parameters as decision points
The calculator converts any quadratic into a(x − h)^2 + k, where (h,k) is the turning point. In practice, h is the input value that optimizes the outcome, and k is the optimized output. When a>0 the vertex is a minimum; when a<0 it is a maximum. These interpretations support quick checks in cost, area, and trajectory problems.
Axis of symmetry and predictable behavior
The axis x=h splits the parabola into mirror halves. That symmetry reduces verification time: values at h±d share the same y. For example, if h=2, then x=1.5 and x=2.5 produce equal outputs. This is useful when building tables, checking graphs, or spotting transcription errors from handwritten work.
Discriminant as a roots indicator
The discriminant Δ=b^2−4ac is reported to summarize intersection behavior. If Δ>0, the curve crosses the x-axis twice; if Δ=0, it touches once; if Δ<0, roots are complex. This single value helps you anticipate whether real solutions exist before substituting numbers into the quadratic formula.
Graph preview for immediate validation
The Plotly preview draws the parabola using the computed standard coefficients. The x-range is centered around the vertex, with a symmetric span for stable viewing. The vertex marker allows instant confirmation that the computed h aligns with the graph’s turning point, while curvature direction matches the sign of a. This prevents common mistakes such as sign flips inside (x−h).
Exports for reporting and study workflows
CSV export produces a compact record of inputs, forms, discriminant, and steps, making it convenient for spreadsheets and learning logs. PDF export captures the highlighted result block, including the vertex form and key values, which is useful for assignments, lab notes, or client-ready documentation. Both exports are generated on-demand after a successful calculation.
Precision handling and rounding strategy
Results are formatted to six decimals and trimmed for readability, balancing precision and clarity. Very small values are normalized to zero to avoid visual noise. When coefficients are large, interpret results using the exact formulas shown in the steps, and treat rounded displays as presentation values. This approach keeps the calculator reliable for typical classroom and applied problem scales. It also supports quick classroom checks and professional reviews.
FAQs
1) What does vertex form tell me quickly?
It shows the turning point directly as (h,k) and the opening direction from a. That makes optimization and graphing faster than reading values from standard form.
2) Why must a be non-zero?
If a=0, the expression is linear, not quadratic. Vertex form and parabola properties like a turning point do not apply.
3) How is h computed from standard form?
The calculator uses h=-b/(2a). This is the x-coordinate where the slope of the parabola is zero, so the function reaches its minimum or maximum there.
4) What if the discriminant is negative?
The parabola does not cross the x-axis, so there are no real roots. The calculator reports complex roots in re ± imi form while still providing the real vertex and axis.
5) Can I convert vertex form back to standard form?
Yes. Choose “Vertex → Standard” and enter a,h,k. The tool expands the square and returns the equivalent ax^2+bx+c with computed b and c.
6) Why might my printed values look rounded?
Displayed numbers are formatted to six decimals for clean reading. The formulas shown explain the exact computation, so you can reproduce the precise values if your work requires higher precision.