Introduction to Vertex form of a parabola

In vertex form $y = a(x - h)^2 + k$, the vertex sits at $(h, k)$.
It is the turning point of the parabola — the lowest point when it opens up, the highest when it opens down.
The constant $a$ decides the direction and the shape.

The figure shows a parabola in vertex form.
The vertex $(h, k)$ is at the bottom here because $a > 0$.
The dashed line through the vertex is the axis of symmetry, the vertical line $x = h$.

Watch the subtraction inside the bracket: $(x - h)$.
A vertex at $x = 3$ gives $(x - 3)$, while a vertex at $x = -1$ gives $(x - (-1)) = (x + 1)$.
So a plus sign inside the bracket means $h$ is negative.

The value of $a$ also controls the shape.
A larger $|a|$ makes the parabola narrower, and a smaller $|a|$ makes it wider.
But it does not move the vertex, which stays at $(h, k)$.

What is the vertex of $y = (x - 3)^2 + 4$?
Compare it to $y = a(x - h)^2 + k$: $h = 3$ and $k = 4$.
So the vertex is $(3, 4)$.

What is the vertex of $y = -2(x + 1)^2 + 3$?
Write $(x + 1)$ as $(x - (-1))$, so $h = -1$ and $k = 3$.
The vertex is $(-1, 3)$, and since $a = -2 < 0$, the parabola opens downward.

Write the vertex form of a parabola with vertex $(4, -3)$.
Substitute $h = 4$ and $k = -3$ into $y = a(x - h)^2 + k$.
This gives $y = a(x - 4)^2 - 3$.