Quadratic Functions & Vertex Form
Explore how parameters in vertex form y = a(x - h)² + k control the shape and position of parabolas
- Understand vertex form and its geometric meaning
- Identify vertex, axis of symmetry, and direction
- Calculate x-intercepts using the discriminant
- Connect completing the square to vertex form
y = a(x - h)² + k
a: stretch/reflection
h: horizontal shift
k: vertical shift
The vertex form of a quadratic is y = a(x - h)² + k.
The vertex (turning point) is at (h, k). The parameter a
controls the width and direction. When a > 0 the parabola opens upward, when a < 0
it opens downward. A larger |a| makes the parabola narrower, a smaller |a| makes it wider.
The axis of symmetry is x = h.
Goal: Explore how a, h, and k shape the parabola. Identify the vertex, axis of symmetry and intercepts.
Real-World: Quadratic functions describe the path of a ball thrown through the air, the shape of satellite dishes, and the curve of suspension bridge cables.
Parameters
Adjust a, h, and k to see how each parameter reshapes the parabola. The vertex form y = a(x − h)² + k makes these effects crystal clear.
Original: y = x²
Your function
Vertex & Axis
Key Features
These properties are read directly from the vertex form. The axis of symmetry always passes through x = h, and the vertex sits at (h, k).
( )
x =