Parabolas as graphs of quadratics, vertex form and standard form, the vertex and axis of symmetry, and the focus and directrix.
The graph of a quadratic
The graph of \(y = ax^2 + bx + c\) (with \(a \neq 0\)) is a parabola: a U-shaped curve with a vertical axis of symmetry. The turning point on the axis is the vertex.
Vertex form and standard form
Finding the equation
Standard form has three coefficients, so three points determine a parabola. With the vertex known, vertex form leaves only \(a\) to find.
Focus and directrix
A parabola has a geometric definition too.
The vertex is halfway between the focus and the directrix. Call \(p\) the directed distance from the vertex to the focus.
Practice
Quick check
What is the vertex of \(y = (x - 4)^2 + 7\)?
In vertex form \(y = a(x - h)^2 + k\), the vertex is \((h, k)\).
Quick check
Where is the focus of \(x^2 = 12y\)?
\(4p = 12\), so \(p = 3\) and the focus is \((0, 3)\).
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