A parabola is the set of all points in a plane that are equidistant from a fixed point (the focus) and a fixed line (the directrix).
The vertex of the parabola is the midpoint between the focus and the directrix.
NoteThe vertexis the point where the parabola changes direction.
Finding the Equation of a Parabola from its Focus and Directrix
The equation of a parabola with vertex $(h, k)$ and focus $(h, k + p)$ (where $p$ is the distance from the vertex to the focus) is:
$$y = a(x - h)^2 + k$$
where $a = \frac{1}{4p}$.
NoteThe vertex formof a parabola is $y = a(x - h)^2 + k$, where $(h, k)$ is the vertex.
To find the equation of a parabola given its focus and directrix:
- Find the vertex: The vertex is the midpoint between the focus and the directrix.
- Calculate \$p\$: The distance from the vertex to the focus (or directrix) is \$p\$.
- Determine \$a\$: Use \$a = \frac{1}{4p}\$.
- Write the equation: Substitute \$h\$, \$k\$, and \$a\$ into the vertex form.
Be careful with the sign of \$p\$. If the parabola opens downwards, \$p\$ is negative.
Converting to Standard Form
The standard form of a parabola is \$y = ax^2 + bx + c\$.
To convert from vertex form to standard form:
- Expand the squared term: \$(x - h)^2 = x^2 - 2hx + h^2\$.
- Distribute \$a\$: \$a(x^2 - 2hx + h^2) = ax^2 - 2ahx + ah^2\$.
- Add \$k\$: \$ax^2 - 2ahx + ah^2 + k\$.
1. Find the equation of the parabola with focus \$(2, 3)\$ and directrix \$y = -1\$. 2. Convert the equation from vertex form to standard form.
Theory of KnowledgeHow does the concept of a parabola relate to the idea of symmetry in mathematics? Why is symmetry such a powerful tool in mathematical reasoning?
