Parabola

A parabola is a symmetric curve produced by a quadratic function and defined geometrically by equal distance from a focus and directrix.

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The vertex anchors a parabola’s position and symmetry. A U-shaped curve is mirrored across a dashed vertical axis. Its highlighted vertex is the turning point, so vertex form and the axis of symmetry can be read from the same picture.

A parabola is the set of points equally distant from a fixed focus and a fixed line called the directrix.

The familiar quadratic graph

Vertical parabolas arise from

$$ y=a(x-h)^2+k. $$

The vertex is $(h,k)$ and the axis of symmetry is $x=h$. The curve mirrors across that axis.

Opening and shape

If $a>0$, the parabola opens upward; if $a<0$, it opens downward. Increasing $|a|$ creates greater vertical stretch, making the curve appear narrower. Values of $|a|$ between $0$ and $1$ create vertical compression.

Reflection and stretch occur around the vertex, not necessarily the origin.

From standard form

For $y=ax^2+bx+c$, axis is $x=-b/(2a)$. Substitute to find the vertex. Completing the square converts to vertex form and makes translations visible.

The discriminant $b^2-4ac$ predicts the number of real $x$-intercepts.

Sketching checklist

Plot vertex, axis, opening, intercepts, and symmetric points. Draw one smooth curve. The arms continue indefinitely and do not become straight.

Common mistakes

Reading vertex signs directly from brackets. In $(x-h)^2$, horizontal shift is $h$.

Calling the graph V-shaped. A parabola is smooth; $y=|x|$ is V-shaped.

Assuming vertical orientation. Parabolas can open left or right.

Forgetting symmetry when plotting. Use point pairs to check the sketch.

Quick self-check

Where are vertex and axis?
What do sign and size of $a$ or $p$ imply?
Are plotted points symmetric?
Is the graph a function in the chosen orientation?