Vertex Form and Parabola Transformations

h is the x-coordinate of the vertex (the turning point). The bracket $(x-h{)}^2$ equals zero when $x=h$, and that is exactly where the parabola turns, so h fixes its horizontal position.

Watch the minus sign built into the form: $y=2(x+3{)}^2-5$ is really $(x-(-3){)}^2$, so $h=-3$, not 3. The letter a controls the shape and which way the parabola opens, and k gives the y-coordinate of the vertex.

k is the y-coordinate of the vertex. When $x=h$ the squared bracket equals 0, so $y=k$, the lowest (or highest) value the curve reaches. Adding k shifts the whole parabola vertically.

k is not the constant term of the expanded quadratic: expanding $y=a(x-h{)}^2+k$ gives $a{x}^2-2ahx+a{h}^2+k$, so the constant is $a{h}^2+k$. The x-coordinate of the vertex is h, and a controls the shape.

Compare $y=-3(x-4{)}^2+7$ with the vertex form $y=a(x-h{)}^2+k$.

The bracket $(x-4)$ is zero when $x=4$, so $h=4$. The number added outside the bracket is $k=7$. The vertex is therefore $(4,7)$.

The $-3$ is the value of a: it makes the parabola narrower and turns it upside down, but its minus sign belongs to a alone and changes neither coordinate of the vertex.

Compare $y=2(x+3{)}^2-5$ with the vertex form $y=a(x-h{)}^2+k$.

Rewrite $(x+3)$ as $(x-(-3))$, so $h=-3$: the bracket is zero when $x=-3$. The number outside the bracket is $k=-5$.

The vertex is therefore $(-3,-5)$. Both coordinates are negative here, so keep the signs.

Compare $y=-4(x-1{)}^2+6$ with the vertex form $y=a(x-h{)}^2+k$.

The bracket $(x-1)$ is zero when $x=1$, so $h=1$, and the number outside the bracket is $k=6$. The vertex is therefore $(1,6)$.

The $-4$ is the value of a: it turns the parabola upside down, but that minus sign belongs to a and does not make $k$ negative.

Because $a=-5$ is negative, the parabola opens downwards, so its vertex is the highest point.

Compare $y=-5(x+5{)}^2-2$ with the vertex form $y=a(x-h{)}^2+k$. Rewrite $(x+5)$ as $(x-(-5))$, so $h=-5$, and the number outside the bracket is $k=-2$.

The highest point is therefore $(-5,-2)$.