The parabola is one of the conic sections, which are the group of curves obtained by intersecting a cone with a plane. A parabola is produced by intersecting the cone with a plane parallel to its generating line. By intersecting the cone at other angles, we can produce circles, hyperbolas, and ellipses.
Definition Parabola
Given a quadratic function \(x \mapsto ax^2 + bx + c\) where \(a \neq 0\), its graph is called a parabola.
For any quadratic function \(x \mapsto ax^2 + bx + c\), \(a \neq 0\):
If \(a > 0\), the graph is concave up: .
If \(a < 0\), the graph is concave down: .
Solving \(f(x) \equal y\)
Method Solving \(f(x) \equal y\)
When solving for a value of \(f(x) = y\), we obtain a quadratic equation in \(x\). Since it is quadratic, there may be 0, 1, or 2 real solutions for \(x\).
Example
For \(f(x) = 2x^2 - 5x + 2\), find the \(x\)-intercepts of the function.