Vertex Form & Transformations
Read y = a(x − h)² + k as a transformation map: a controls opening and width, while h and k place the turning point.
How can three parameters place and shape an entire parabola?
Vertex form starts with the parent curve y = x², then stretches, reflects, and translates it without hiding the turning point.
The vertex is (h, k), and the axis of symmetry is x = h.
Read y = 2(x − 1)² − 3 from the vertex outward.
The curve turns at (1, −3), and its mirror line is x = 1.
Positive a opens upward, so the vertex is a minimum.
The output changes faster than y = x², so the parabola appears narrower.
The sign controls opening. Magnitude controls vertical stretch or compression.
The sign appears opposite inside the parentheses: x + 3 means h = −3.
k raises or lowers every output, including the vertex.
Change one parameter at a time.
Holding the other parameters fixed makes each job visible instead of blending three effects into one slider cloud.