2.2.2 Parabola Properties and Graphs¶

Analyze the key features of parabolas such as vertex, axis of symmetry, focus, directrix, and direction of opening.

定义¶

A parabola is the set of all points in a plane that are equidistant from a fixed point called the focus and a fixed line called the directrix. The parabola has several key features: (1) The vertex is the point on the parabola closest to the directrix, located midway between the focus and directrix. (2) The axis of symmetry is a vertical or horizontal line passing through the vertex and focus, about which the parabola is symmetric. (3) The focus is a fixed point at distance $p$ from the vertex along the axis of symmetry. (4) The directrix is a fixed line at distance $p$ from the vertex, perpendicular to the axis of symmetry. (5) The parameter $p$ determines the "width" or "openness" of the parabola. For a parabola with vertex at $(h, k)$ and vertical axis of symmetry, the standard form is $(x-h)^2 = 4p(y-k)$, where the parabola opens upward if $p > 0$ and downward if $p < 0$. For a horizontal axis of symmetry, the form is $(y-k)^2 = 4p(x-h)$, opening rightward if $p > 0$ and leftward if $p < 0$.

核心公式¶

$(x-h)^2 = 4p(y-k)$ (vertical axis of symmetry, vertex at $(h,k)$)
$(y-k)^2 = 4p(x-h)$ (horizontal axis of symmetry, vertex at $(h,k)$)
$y = ax^2 + bx + c$ (standard form), where vertex is at $\left(-\frac{b}{2a}, c - \frac{b^2}{4a}\right)$ and $p = \frac{1}{4a}$
$Focus: $(h, k+p)$ for vertical parabola; $(h+p, k)$ for horizontal parabola$
$Directrix: $y = k - p$ for vertical parabola; $x = h - p$ for horizontal parabola$

易错点¶

⚠️ Confusing the direction of opening: students often forget that when $p > 0$, a vertical parabola opens upward (not downward), and the focus is above the vertex. Similarly, for horizontal parabolas, $p > 0$ means opening rightward.
⚠️ Incorrectly calculating the vertex from standard form $y = ax^2 + bx + c$: students may forget the negative sign in the x-coordinate formula $-\frac{b}{2a}$, or miscalculate the y-coordinate by not properly substituting back into the equation.
⚠️ Misidentifying the relationship between the parameter $p$ and the coefficient $a$ in $y = ax^2 + bx + c$: students forget that $a = \frac{1}{4p}$, so larger $|a|$ means smaller $|p|$ and a narrower parabola, not wider.
⚠️ Placing the focus and directrix on the wrong side of the vertex: the focus should always be between the vertex and the direction of opening, while the directrix is on the opposite side of the vertex from the focus.