Parabola

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

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.

Vertex and symmetry

The vertex is the turning point and lies halfway between symmetric points. If $(h-d,y)$ is on a vertical parabola, then $(h+d,y)$ is also on it.

The minimum value is $-5$ because the graph opens upward.

Intercepts

Set $x=0$ for the $y$-intercept and $y=0$ for $x$-intercepts. A vertical parabola may cross the $x$-axis twice, touch once at the vertex, or not cross.

For $y=(x-1)(x-5)$, zeros are $1$ and $5$, and the symmetry axis lies at their midpoint $x=3$.

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.

Focus and directrix

For

focus is $(h,k+p)$ and directrix is $y=k-p$. The sign of $p$ gives opening direction. The reflective property following from equal distances explains parabolic satellite dishes, headlights, and microphones.

Horizontal parabolas

A horizontal parabola has form

It may fail the vertical-line test because one $x$ can correspond to two $y$-values. It is still a parabola, though not $y$ as a function of $x$ over the whole curve.

Sketching checklist

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

Applications

Projectile height over time, bridge arches, reflector design, and some optimization models use parabolas. A real model may use only part of the mathematical curve because time, length, or height is restricted.

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?

Related topics

• Quadratic Function
• Vertex
• Axis of Symmetry
• Conic Sections