EASA / CAR Part-66 · Module 1 · 1.2 Algebra · Cat B1/B2

Quadratic Equations & the Parabola

Objective: a squared term bends the graph into a parabola. Change a, b and c and watch the curve move, while the discriminant tells you in advance whether it will cross the axis twice, once, or not at all.

y = ax² + bx + c

a 1.0

b 0

c -4

Discriminant b²−4ac

Root 1

Root 2

Vertex x

x = (−b ± √(b² − 4ac)) / 2a · vertex at x = −b / 2a

The discriminant b² − 4ac decides everything before you solve. Positive means the square root is a real number and you get two roots; zero means the ± adds nothing and the two roots merge into one, with the curve just touching the axis; negative means no real roots at all and the curve misses the axis entirely.

a controls the shape, c controls the height. A positive a opens the parabola upward and a negative a flips it over, while a larger size of a makes it narrower. c is simply where the curve cuts the y axis, since putting x = 0 leaves y = c.

The vertex always sits at x = −b/2a, exactly halfway between the two roots because a parabola is symmetrical. If you know one root and the vertex, the other root is the same distance the other side.

Try this: Start with a = 1, b = 0, c = −4. Read the roots. Can you factorise x² − 4 to check?
Try this: Raise c until the discriminant passes through zero. What is the curve doing at that exact moment?
Try this: Make a negative. Which way does the parabola open, and where is the vertex now?