The U-Shape: Parabolas
Explore how a, b, and c shape the graph of a quadratic equation.
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
- Make the parabola open downwards.
- Put the vertex on the y-axis: b = 0.
- Make the graph touch the x-axis at exactly one point: slide c to 0 so D = b² − 4ac = 0.
- Open the parabola upwards and twice as steep: set a = 2, and set b = 4 to shift the vertex left.
- Now make it TOUCH the x-axis exactly once. The discriminant b² − 4ac must be 0, so with b = 4 and a = 2 the constant c has to be 2.
Check yourself
Which coefficient determines whether the parabola opens up or down?
If the vertex of the parabola is at (0, 0), what must be true about b and c?
- a, the coefficient of x^2 — correct
- b, the coefficient of x
- c, the constant term
- The discriminant (b^2 - 4ac)
- b = 0 and c = 0 — correct
- a = 0 and b = 0
- b = c
- a must be negative
Think about it
- Before you move the slider, predict: what happens to the curve if a becomes negative?
- Predict: if you increase c from 0 to 5, which way does the graph move?