From standard form to vertex form.

Step 1 of 3

Where the vertex is

The vertex sits on the axis of symmetry, halfway between the two roots. So its $x$⁠-coordinate is the average of the roots, and the quadratic formula gives:

$x = -\frac{b}{2a}$⁠

The formula works even when there are no roots at all. You get the vertex’s $y$⁠ by substituting. The Scene shows $y = 2x^2 - 8x + 3$⁠, with the vertex left unmarked.

Step 2 of 3

Completing the square

From the vertex you can get to vertex form. For $y = 2x^2 - 8x + 3$⁠:

$y = 2(x^2 - 4x) + 3 = 2\big((x-2)^2 - 4\big) + 3 = 2(x-2)^2 - 5$$y =$$2(x^2 - 4x) + 3 =$$2\big((x-2)^2 - 4\big) + 3 =$$2(x-2)^2 - 5$⁠

Now the vertex and the axis are marked. Step through Look inside: a stretch by $2$⁠, a shift of $2$⁠ to the right and a shift of $5$⁠ down build the parabola from $y = x^2$⁠, exactly as vertex form says.

Step 3 of 3

Touch the axis once

a and c are locked, and the Scene shows $\Delta$⁠ and the roots. Only b moves.