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.
y=0.5x² + x - 2.5Δ 6Roots 2 real: -3.45, 1.45
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?