The General Equation of a Translated Parabola
Simple Explanation
When a parabola's vertex is NOT at the origin, but instead at some point (h,k), its equation becomes (x−h)² = 4p(y−k) — the standard-form equation, directly translated.
Why Do We Need It?
Most real parabolas encountered in applications are not conveniently centered at the origin — this is the general form actually needed to describe them.
See It
A parabola opening upward, with its vertex marked at the point (2,3), shifted away from the origin
Formula
The General Equation of a Translated Parabola
(x−h)² = 4p(y−k), vertex at (h,k)
A parabola whose vertex has been moved away from the origin, to the point (h,k) — the direct translation of the standard-form equation.
- (h,k)
- — the vertex of the parabola
- p
- — the distance from the vertex to the focus
When to use it: Whenever a parabola's vertex is NOT at the origin.
Worked Example
Write the equation of a translated parabola
Write the equation of a parabola with vertex (2,3) and p=1 (opening upward).
Why Does This Work?
Shifting a curve h units right and k units up is achieved by replacing x with (x−h) and y with (y−k) everywhere in the original equation — exactly the same translation technique used for circles, applied here to the standard parabola equation.
Real-Life Example
Positioning a suspension bridge's cable curve
A suspension bridge's main cable follows a parabolic curve, but its lowest point (vertex) is at the bridge's specific height and horizontal position, not at the origin.
The translated parabola equation describes the cable's actual real-world shape and position directly.
Practice
A parabola (x−1)²=8(y+2) has vertex (1,k). Find k.
MediumCommon mistake
Forgetting that a translated parabola's focus and directrix also shift along with the vertex — they are no longer at (0,p) and y=−p, but at (h,k+p) and y=k−p respectively.
Quick Review
- (x−h)² = 4p(y−k), vertex at (h,k).
- Replace x with (x−h) and y with (y−k) to translate the standard equation.
- The focus and directrix shift along with the vertex.