Read a quadratic from its shape
When does a curved model reach zero?
Entering midway? Squaring measures a nonnegative product x×x. A square root gives a nonnegative number whose square matches the input.
Start with a concrete question
Explore a positive-opening quadratic in vertex form. The same expression can be expanded into ax²+bx+c to connect algebra and geometry.
Try the initial inputs, then write a prediction before changing one setting. All calculations run locally on your CPU.
Go directly to the experiment · Review prerequisite mathematics and return here
What do you expect?
Predict how moving the vertex above zero changes real roots.
Change one thing. Inspect what happens.
- At defaults locate the vertex and calculate both roots.
- Raise k to zero, then above zero.
- Increase a with h,k fixed; compare root spacing and curvature.
Your settingsMechanism
Calculation belowEvidence
Compare outcomes
Quadratic and zero level
Horizontal: x. Vertical: y.
- 1. Quadratic (solid line)
- 2. Zero level (dashed line)
Axes use the displayed ranges.
Read plotted values as a table
| Series | x | y |
|---|---|---|
| Quadratic | -6 | 45 |
| Quadratic | -5.75 | 41.56 |
| Quadratic | -5.5 | 38.25 |
| Quadratic | -5.25 | 35.06 |
| Quadratic | -5 | 32 |
| Quadratic | -4.75 | 29.06 |
| Quadratic | -4.5 | 26.25 |
| Quadratic | -4.25 | 23.56 |
| Quadratic | -4 | 21 |
| Quadratic | -3.75 | 18.56 |
| Quadratic | -3.5 | 16.25 |
| Quadratic | -3.25 | 14.06 |
| Quadratic | -3 | 12 |
| Quadratic | -2.75 | 10.06 |
| Quadratic | -2.5 | 8.25 |
| Quadratic | -2.25 | 6.563 |
| Quadratic | -2 | 5 |
| Quadratic | -1.75 | 3.563 |
| Quadratic | -1.5 | 2.25 |
| Quadratic | -1.25 | 1.063 |
| Quadratic | -1 | 0 |
| Quadratic | -0.75 | -0.9375 |
| Quadratic | -0.5 | -1.75 |
| Quadratic | -0.25 | -2.438 |
| Quadratic | 0 | -3 |
| Quadratic | 0.25 | -3.438 |
| Quadratic | 0.5 | -3.75 |
| Quadratic | 0.75 | -3.938 |
| Quadratic | 1 | -4 |
| Quadratic | 1.25 | -3.938 |
| Quadratic | 1.5 | -3.75 |
| Quadratic | 1.75 | -3.438 |
| Quadratic | 2 | -3 |
| Quadratic | 2.25 | -2.438 |
| Quadratic | 2.5 | -1.75 |
| Quadratic | 2.75 | -0.9375 |
| Quadratic | 3 | 0 |
| Quadratic | 3.25 | 1.063 |
| Quadratic | 3.5 | 2.25 |
| Quadratic | 3.75 | 3.563 |
| Quadratic | 4 | 5 |
| Quadratic | 4.25 | 6.563 |
| Quadratic | 4.5 | 8.25 |
| Quadratic | 4.75 | 10.06 |
| Quadratic | 5 | 12 |
| Quadratic | 5.25 | 14.06 |
| Quadratic | 5.5 | 16.25 |
| Quadratic | 5.75 | 18.56 |
| Quadratic | 6 | 21 |
| Zero level | -6 | 0 |
| Zero level | 6 | 0 |
Follow the calculation
- Rule y = 1(x − 1)² + (-4). Vertex = (1, -4).
- Expanded coefficients: 1x² + (-2)x + (-3).
- For y=0: (x − 1)² = 4.
- Roots x = 1 ± √(4) = -1, 3.
The vertex form exposes the turning point. A quadratic may cross zero twice, touch once or never reach zero in the real numbers.
Captures keep your inputs, prediction and observation. Results are recalculated from those inputs. Resetting inputs keeps your captures and challenge history.
Explain the mechanism
Read a quadratic from its shape
Recap for entering here
Squaring measures a nonnegative product x×x. A square root gives a nonnegative number whose square matches the input.
The concrete question
Explore a positive-opening quadratic in vertex form. The same expression can be expanded into ax²+bx+c to connect algebra and geometry.
Predict how moving the vertex above zero changes real roots.
Work through the mechanism
The square is smallest at x=h. For a>0, the minimum output is k, so the vertex is (h,k).
At defaults y=(x−1)²−4. Set y=0: (x−1)²=4, giving x−1=±2, so roots are −1 and 3.
Expansion gives ax²−2ahx+ah²+k. Completing the square reverses this transformation and explains how a general quadratic connects to its vertex.
Formula and symbols
y=a(x−h)²+k, a>0. Real roots, when k≤0, are x=h±√(−k/a).
Test the idea
At defaults locate the vertex and calculate both roots.
Raise k to zero, then above zero.
Increase a with h,k fixed; compare root spacing and curvature.
Assumptions and boundaries
This investigation uses positive curvature; downward-opening quadratics reverse minimum to maximum.
Negative square roots have no real value; complex-number methods belong to further study.
Extend your investigation
High school: Compare a parabola’s symmetry with its table of values.
College: Derive the quadratic formula by completing the square.
Working professionals: Explain the valid domain of a quadratic cost or trajectory approximation.
Try a fresh case
For y=2(x−3)²−8, what is the larger real root?
This is a public practice check. Retries support learning; matching a number does not certify independent proficiency.
Take the investigation further
Choose any extension. These are project prompts for you to carry out; they are not automatically assessed.
School investigation
Compare a parabola’s symmetry with its table of values.
College investigation
Derive the quadratic formula by completing the square.
Workplace investigation
Explain the valid domain of a quadratic cost or trajectory approximation.
Keep the evidence, choose your next step
Completion records your own decision that you have worked through this investigation. Practice checks and independent project evidence are separate.