Parabola
A parabola is a curve you’ve already seen many times before. It’s the shape of the graph of a quadratic function.
However, when we talk about parabolas in the context of conic sections, we
usually mean an equation in the form
Equation of a parabola
As mentioned before, the equation of a right-opening parabola is:
y^2 = 4ax
a is the distance from the vertex to the focus (the point where the parabola is “pointiest”).
Examples
Find the equation of the parabola with focus at (2, 0)
- The distance from the vertex to the focus is 2, so
a = 2 . y^2 = 4 \cdot 2 \cdot x = 8x .
Find the equation of the parabola with focus at (5, 0)
- The distance from the vertex to the focus is 5, so
a = 5 . y^2 = 4 \cdot 5 \cdot x = 20x .
Find the equation of the parabola with focus at (0, 0)
- The distance from the vertex to the focus is 0, so
a = 0 . y^2 = 4 \cdot 0 \cdot x = 0 .
Find the focus of the parabola with equation y^2 = 12x
4a = 12 \Rightarrow a = 3 .- The focus is at (3, 0).
Find the focus of the parabola with equation y^2 = 32x
4a = 32 \Rightarrow a = 8 .- The focus is at (8, 0).
Find the focus of the parabola with equation y^2 = 0
4a = 0 \Rightarrow a = 0 .- The focus is at (0, 0).