Integrated Math 2 / Unit 3: Quadratic Functions
3.2 Vertex Form
Notes
Vertex form is $f(x) = a(x - h)^2 + k$. The vertex is $(h, k)$ and the axis is $x = h$.
- Watch the sign of $h$: $y = (x + 3)^2 - 8$ has $h = -3$, so the vertex is $(-3, -8)$.
- If $a > 0$, $k$ is the minimum value. If $a < 0$, $k$ is the maximum value.
- To write the equation from the vertex and one more point, substitute the point to find $a$.
- Example: vertex $(2, -1)$ through $(3, 1)$: $1 = a(1)^2 - 1$, so $a = 2$ and $y = 2(x - 2)^2 - 1$.
Practice
What is the vertex of $y = (x + 3)^2 - 8$?
Write it as $(x - (-3))^2 + (-8)$, so $h = -3$ and $k = -8$.
What is the maximum value of $y = -2(x - 1)^2 + 5$?
The parabola opens down ($a = -2$), so the vertex value $k = 5$ is the maximum.
A parabola has vertex $(2, -1)$ and passes through $(3, 1)$. Which equation matches?
Substitute: $1 = a(3 - 2)^2 - 1$ gives $a = 2$.
Ready for more? Take the Unit 3 test.