What quantity did you change?
Labs / Mathematics
Quadratic Explorer
Drag the vertex around the grid, or nudge the sliders for a, h, and k. The curve redraws instantly in vertex form y = a(x − h)² + k, with the axis of symmetry and x-intercepts marked. Flip to standard form to see the same parabola expanded, and watch how the discriminant predicts the number of real roots.
Labs / Mathematics / Quadratic Explorer
Quadratic Explorer
Drag the vertex around the grid, or nudge the sliders for a, h, and k. The curve redraws instantly in vertex form y = a(x − h)² + k, with the axis of symmetry and x-intercepts marked. Flip to standard form to see the same parabola expanded, and watch how the discriminant predicts the number of real roots.
Why it matters
Why explore this?
Mathematics helps us describe patterns precisely and predict what comes next. This model turns an abstract relationship into something you can change and observe.
Explore
Try the model
y=0.5x² + x - 2.5Δ 6Roots 2 real: -3.45, 1.45
Investigate
Questions to test
Predict first, try it in the model, then use what you observe to answer.
What to try
- Slide the vertex above the x-axis while a stays positive — where did the roots go, and why does the discriminant turn negative?
- What has to be true about the signs of a and k for a parabola to cross the x-axis twice?
- Can you land the vertex exactly on the x-axis? What does the discriminant read at that moment, and how many roots are there?
- Make a very small, then very large. How does a control the width, and does it move the roots?
- Switch to standard form and drag h. Why do both b and c change even though you only touched the horizontal shift?
Explain
Make sense of what happened
State the pattern in words, test it with another example, and then connect it to the mathematical rule or representation shown in the model.
Check
Before you leave
What pattern did you observe?
Does the pattern work for another example?
If you can answer all three using evidence from the model, you understand the main idea. If not, return to Explore and change one setting at a time.