20.3: Parabolas
- Page ID
- 174401
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Definitions and Theorems
A parabola is the set of all points in a plane that are the same distance from a fixed line, called the directrix, and a fixed point called the focus (where the focus is not on the directrix).
The line segment that passes through the focus and is parallel to the directrix is called the latus rectum.
The table below and the following figure summarize the standard features of parabolas with a vertex at the origin.
| Axis of Symmetry | Equation | Focus | Directrix | Endpoints of Latus Rectum |
|---|---|---|---|---|
| \( x \)-axis | \( y^2 = 4px \) | \( (p,0) \) | \( x = -p \) | \( (p, \pm 2p) \) |
| \( y \)-axis | \( x^2 = 4py \) | \( (0,p) \) | \( y = -p \) | \( (\pm 2p, p) \) |
(b) When \( p < 0 \) and the axis of symmetry is the \( x \)-axis, the parabola opens left.
(c) When \( p > 0 \) and the axis of symmetry is the \( y \)-axis, the parabola opens up.
(d) When \( p < 0 \) and the axis of symmetry is the \( y \)-axis, the parabola opens down.
The table below and the following figure summarize the standard features of parabolas with a vertex at the origin.
| Axis of Symmetry | Equation | Focus | Directrix | Endpoints of Latus Rectum |
|---|---|---|---|---|
| \( y = k \) | \( (y - k)^2 = 4p(x - h) \) | \( (h + p,k) \) | \( x = h-p \) | \( (h+p, k \pm 2p) \) |
| \( x = h \) | \( (x - h)^2 = 4p(y - k) \) | \( (h,k+p) \) | \( y = k-p \) | \( (h \pm 2p, k + p) \) |
(b) When \( p < 0 \), the parabola opens left.
(c) When \( p > 0 \), the parabola opens up.
(d) When \( p < 0 \), the parabola opens down.