Calculator Input Panel
Example Data Table
Example equation: y = x2 - 4x + 3
| x | y |
|---|---|
| -1 | 8 |
| 0 | 3 |
| 1 | 0 |
| 2 | -1 |
| 3 | 0 |
| 4 | 3 |
| 5 | 8 |
Formula Used
Standard equation: y = ax2 + bx + c, where a ≠ 0.
Discriminant: D = b2 - 4ac. It determines whether the roots are real, repeated, or complex.
Roots: x = (-b ± √D) / 2a. Real roots appear where the graph crosses the x-axis.
Vertex: h = -b / 2a and k = f(h). The vertex is the turning point of the parabola.
Axis of symmetry: x = -b / 2a. The graph mirrors across this vertical line.
Y-intercept: When x = 0, y = c. This gives the graph’s crossing on the y-axis.
Focus and directrix: Focus is (h, k + 1/4a) and directrix is y = k - 1/4a.
How to Use This Calculator
- Enter the quadratic coefficients a, b, and c.
- Set the minimum x, maximum x, and step size.
- Choose the number of decimal places for displayed values.
- Click Plot Quadratic Graph to calculate the equation features.
- Review the roots, vertex, symmetry line, focus, and directrix.
- Study the interactive Plotly graph and point table preview.
- Download the full data table as CSV when needed.
- Export a compact PDF summary for reports or classroom use.
Frequently Asked Questions
1. What does coefficient a change on the graph?
Coefficient a controls opening direction and vertical stretch. Positive values open upward. Negative values open downward. Larger absolute values make the parabola narrower, while smaller absolute values make it wider.
2. Why are some quadratic roots complex?
Complex roots appear when the discriminant is negative. In that case, the parabola never crosses the x-axis. The calculator still reports the roots using real and imaginary parts.
3. What is the vertex of a quadratic graph?
The vertex is the turning point of the parabola. It is the minimum point when a is positive and the maximum point when a is negative.
4. Why should I adjust the x range?
Changing the x range helps you see the most useful part of the curve. A wider range shows more context, while a tighter range highlights the vertex and intercept behavior.
5. What does the step size affect?
Step size controls how many x values are sampled. Smaller steps create smoother graphs and larger data tables. Bigger steps calculate faster but may miss detail around the turning point.
6. Can this calculator help in teaching algebra?
Yes. It connects the symbolic equation with a plotted parabola, intercepts, symmetry, and data points. That makes classroom explanation and homework checking much easier.
7. What is the axis of symmetry?
The axis of symmetry is the vertical line through the vertex. It divides the parabola into two mirrored halves and always has equation x = -b / 2a.
8. Why include CSV and PDF downloads?
CSV files help with spreadsheet analysis and further plotting. PDF export creates a neat summary for assignments, reports, revision notes, or quick sharing with students and colleagues.