Definition, vocabulary and graphic approach
Parabolas naturally occur as the graphs of quadratic functions such as f\left(x\right)=x^2.
Parabola
A parabola is the set of points in the plane whose distance from a point, called the focus of the parabola, is equal to the distance to a line, called the directrix.
Properties of parabolas
Vertex and focus of a parabola
Vertex of a parabola
The line through the focus of a parabola which intersects the directrix of the parabola at 90^\circ intersects the parabola at one point. That point is called the vertex of the parabola and is denoted V.
Axis of symmetry of a parabola
Axis of symmetry
The line passing through the foci of a parabola and perpendicular to the directrix of the parabola is the axis of symmetry of parabola.
A parabola is unchanged if it is reflected across its axis of symmetry.
Equation of parabolas
Equation of a parabola
Consider a quadratic function:
f\left(x\right)=ax^2+bx+c
With a\not=0. Let d=\dfrac{-b}{2a}.
The graph of f\left(x\right) is a parabola with the following properties:
- The vertex of the parabola is at \left(d, f\left(d\right)\right).
- The focus of the parabola is at \left(d, f\left(d\right)+\dfrac{1}{4a}\right).
- The axis of symmetry is the vertical line x=d.
- The directrix of the parabola is the horizontal line y= f\left(d\right)-\dfrac{1}{4a}.
Consider the following quadratic function:
f\left(x\right)=x^2-2x+3
The graph of the function f\left(x\right) is a parabola with the following properties:
- The vertex of the parabola is at \left(\dfrac{-\left(-2\right)}{2},\dfrac{4\left(3\right)-\left(-2\right)^2}{4}\right)=\left(1, 2\right).
- The focus of the parabola is \left(\dfrac{-\left(-2\right)}{2},\dfrac{4\left(3\right)-\left(-2\right)^2+1}{4}\right)=\left(1, \dfrac{9}{4}\right).
- The axis of symmetry of the parabola is the vertical line x=\dfrac{-\left(-2\right)}{2}=1.
- The directrix of the parabola is the horizontal line y=\dfrac{4\left(3\right)-\left(-2\right)^2-1}{4}=\dfrac{7}{4}.
The vertex form of a quadratic equation is f\left(x\right)=a\left(x-h\right)^2+k. When a quadratic function is in vertex form, then:
- The vertex of the graph is \left(h,k\right).
- The focus of the parabola is at \left(h,k+\dfrac{1}{4a}\right).
- The axis of symmetry is x=h.
- The directrix of the parabola is y=k-\dfrac{1}{4a}.
Consider the following quadratic in vertex form:
f\left(x\right)=\left(x-1\right)^2+3
Then:
- The vertex of the graph is at \left(1{,}3\right).
- The focus of the graph is at \left(1{,}3+\dfrac{1}{4}\right)=\left(1, \dfrac{13}{4}\right).
- The axis of symmetry of the graph is x=1.
- The directrix of the graph is y=3-\dfrac{1}{4}=\dfrac{11}{4}.