This tool turns the numbers a, b and c of a quadratic into its discriminant, roots, turning point and a labelled graph. It helps students who can follow the algebra in class but lose the link between an equation and its picture.
Free tool
Quadratic graph and roots explorer
You can factorise a quadratic but cannot see what its roots mean on the graph.
Everything you enter stays on this device.
How do I use it?
- Type a, b and c for the equation y = ax² + bx + c. For x² − 5x + 6, enter 1, -5 and 6.
- Press Explore. The result appears below the boxes.
- Read the discriminant line first, then the roots, then the turning point.
- Look at the graph and find each labelled point: the roots on the x-axis, the turning point and the dashed axis of symmetry.
- Press Reset to return to the starting example, or change one number to see how the graph moves.
How does the tool work?
Every line of the result follows the fixed rules below and shows its working.
| Quantity | Rule used |
|---|---|
| Discriminant D | b² − 4ac |
| Roots | x = (−b ± √D) ÷ 2a, only when D ≥ 0 |
| Axis of symmetry | x = −b ÷ 2a |
| Turning point | x = −b ÷ 2a, then y = c − b² ÷ 4a |
| y-intercept | (0, c) |
When D is positive there are two roots. When D is zero there is one repeated root and the curve touches the x-axis. When D is negative there is no real root, so the tool says so instead of showing a number.
If a is positive the curve opens upward and the turning point is a minimum. If a is negative it opens downward and the turning point is a maximum.
What does a worked example look like?
Enter a = 1, b = −5 and c = 6. The discriminant is (−5)² − 4(1)(6) = 25 − 24 = 1, which is positive.
The roots are (5 ± 1) ÷ 2, so x = 2 and x = 3. The axis is x = 2.5, and the turning point is (2.5, −0.25), a minimum because a is positive.
Now change c to 10. The discriminant becomes 25 − 40 = −15, so the tool reports no real roots and the curve stays above the x-axis.
What can the tool not tell you?
It explores one quadratic at a time. It cannot choose a method for you, mark a written answer or cover every question type in a paper.
It does not solve quadratic inequalities, and it does not teach factorising. A correct graph is also not a complete exam answer, because a paper asks for working, units and a clear statement.
How does it connect to the lessons?
Start with recognising a quadratic from its highest power, then solve factorisable quadratic equations. Use the tool to check that your roots match the points on the graph.
The lesson on finding roots from a quadratic graph uses the same picture. For Additional Mathematics, see using the discriminant to classify roots and solving quadratic inequalities using intervals.
When you want a teacher to go through the gaps with you, read about Mathematics tuition or Additional Mathematics tuition. A one-hour trial class (from RM50) is a real taught lesson.
Common questions
Does the tool work for Mathematics and Additional Mathematics?
Yes. The roots, turning point and graph shape are the same ideas in both subjects. Additional Mathematics adds work such as the discriminant and inequalities, and the tool shows the discriminant so you can practise classifying roots.
What happens if I enter a = 0?
The tool tells you it is no longer a quadratic and shows the straight line y = bx + c with its root. A quadratic needs a non-zero x² term, so the graph would otherwise be a line rather than a curve.
Does the tool store my numbers?
No. The numbers stay on your device and nothing is sent anywhere. There is no account and nothing to save, so a shared phone or computer is fine.
Can I use it to check my homework answers?
Yes, as a check on your own working. Enter the coefficients, compare each step, and look for the first line where your work and the tool differ.
If the graph and the algebra still feel like two separate topics, a teacher in a one-to-one lesson can connect them using your own working and go at your pace.
- Online one-to-one lessons for your child with an experienced teacher.
- Your first class is a one-hour trial, from RM50. The fee is agreed before you book.
- Happy with the teacher? Continue with lessons of about 1.5 hours. If not, ask for another teacher.