Shifting a parabola.

Step 1 of 4

Three steps from the basic parabola

You can build any parabola from $y = x^2$⁠ in three steps, and each one goes straight into vertex form:

$y = a(x - p)^2 + k$⁠

1. Stretch by a factor of $a$⁠: the parabola gets narrower or wider, and may flip over.
2. Sideways, a shift of $p$⁠: to the right when $p$⁠ is positive, to the left when $p$⁠ is negative.
3. Up or down, a shift of $k$⁠: up when $k$⁠ is positive, down when $k$⁠ is negative.

The Scene shows $y = 2(x-3)^2 - 2$⁠, with Look inside at the start. Step through the stages and watch the pink vertex travel from $(0, 0)$⁠ to $(3, -2)$⁠.

Step 2 of 4

Which way?

The vertical shift is easy to read: $+k$⁠ lifts the graph. The horizontal shift is the confusing one, because it is written inside the brackets with a minus sign. Try Horizontal shift p in the Scene, then answer.

Step 3 of 4

Hit the ring

There is a ring in the Scene at $(2, -3)$⁠. a is locked at $1$⁠, so only the shifts change the parabola.

Step 4 of 4

Read the vertex

In vertex form you can read the vertex without any working: it is the point $(p, k)$⁠. Just be careful with the sign inside the brackets: $(x+1)^2$⁠ is $(x - (-1))^2$⁠.