The shape and sign of a.

Step 1 of 4

The basic parabola

The Scene shows the graph of $y = x^2$⁠. Every $x$⁠ is squared, so $x = 2$⁠ and $x = -2$⁠ give the same $y = 4$⁠: the graph is symmetric. The dashed purple line is the axis of symmetry, and the pink dot is the vertex, the point where the parabola stops falling and starts to rise.

Now add a coefficient: $y = ax^2$⁠. Move a and watch how the parabola changes. Choose y = x² under Look inside, then y = ax² again, to see how it is built from $y = x^2$⁠ by stretching.

Step 2 of 4

Through zero

a is locked at $1$⁠. Before you move it, predict what will happen as it passes through zero into the negative numbers.

Step 3 of 4

Up or down

When the coefficient $a$⁠ is positive, the parabola opens upward and the vertex is a minimum point. When it is negative, the parabola opens downward and the vertex is a maximum point.

A full quadratic function has more terms: $y = ax^2 + bx + c$⁠. Do they affect the direction too?

Step 4 of 4

Narrow or wide

The sign of $a$⁠ sets the direction, and its size, $|a|$⁠, sets the width. Try a few values of a between $0$⁠ and $3$⁠, and compare how quickly the arms climb.