Why Do Parabolas Look Like That?

Every quadratic function produces a parabola — that familiar U-shape. But why does changing a single number reshape the entire curve? Let’s find out by breaking one apart.

Start Simple

Here’s the simplest quadratic: y = x²

Every x gets squared — negative inputs become positive, so the curve is symmetric. The vertex sits at the origin (0, 0).

What Does Each Number Do?

The general form is y = ax² + bx + c. Use the sliders to change each coefficient and watch the parabola respond:

The Three Coefficients, Decoded

a — The Stretch Factor

• |a| > 1: The parabola gets narrower (steeper sides)
• |a| < 1: The parabola gets wider (flatter)
• a > 0: Opens upward (U shape)
• a < 0: Opens downward (∩ shape)
• a = 0: Not a parabola anymore — it’s a straight line!

b — The Tilt

This one is tricky. It moves the vertex both horizontally and vertically along a hidden path. The vertex x-coordinate is always at x = -b / 2a.

c — The Vertical Shift

The simplest one: c just moves the whole parabola up or down. It’s the y-intercept — the point where the curve crosses the y-axis.

Finding Roots: Where Does the Parabola Cross Zero?

The roots (or zeros) are the x-values where y = 0 — visually, where the parabola crosses the x-axis.

Slide c and watch:

• c < 0: Two roots — the parabola dips below the x-axis
• c = 0: Exactly one root — it just touches the axis
• c > 0: No real roots — the parabola floats above

Comparing Parabolas

Here are three parabolas with different a values, all passing through the origin. Notice how a controls the “width”:

The larger |a| is, the steeper the sides. Think of it as how “quickly” the function grows away from the vertex.