The Shape of a Quadratic
Slide the sliders to see how a, b, and c sculpt the parabola. Find the hidden roots.
Equation
y = 1*x^2 + 0*x + 0
Graph
Table
| x | y |
|---|---|
| -5 | 25 |
| -4 | 16 |
| -3 | 9 |
| -2 | 4 |
| -1 | 1 |
| 0 | 0 |
| 1 | 1 |
| 2 | 4 |
| 3 | 9 |
| 4 | 16 |
| 5 | 25 |
What this lesson covers
What you do
You shape the function y = a*x^2 + b*x + c and watch the graph answer.
Challenges to clear
- Set a = 1 and c = -8. Slide b so the parabola crosses the x-axis at x = 2.
- Roots come in pairs: with that b, the curve ALSO crosses at x = −4 (x² + 2x − 8 = (x−2)(x+4)).
- Build x² + bx − 6: set a = 1 and c = −6.
- Now slide b until the curve crosses the x-axis at x = 1. Roots come in pairs — look left and you will find −6 waiting.
Check yourself
In Goal 2, the vertex touched the x-axis. What does this tell us about the discriminant (b^2 - 4ac)?
- It is positive, so there are two distinct roots.
- It is zero, so there is exactly one real root. — correct
- It is negative, so there are no real roots.
- It equals 'a', so the shape determines the roots.
Think about it
- What happens to the curve's width and direction if you slide a?
- Where does the curve hit the y-axis? Try sliding c.