How to Find the Focus, Vertex, and Directrix of a Parabola?

A parabola consists of three parts: Vertex, Focus, and Directrix. The vertex of a parabola is the maximum or minimum of the parabola and the focus of a parabola is a fixed point that lies inside the parabola. The directrix is outside of the parabola and parallel to the axis of the parabola.

Related Topic

• How to Write the Equation of Parabola

Step-by-Step Guide to Finding the Focus, Vertex, and Directrix of a Parabola

• For a Parabola in the form \(y=ax^2+bx+c\):

Vertex: \((\frac{-b}{2a} , \frac{4ac-b^2}{4a})\), Focus: \((\frac{-b}{2a} , \frac{4ac-b^2+1}{4a})\), Direcrix: \(y=c-(b^2+1)4a\).

Finding the Focus, Vertex, and Directrix of a parabola – Example 1:

Find the vertex and focus of this parabola: \(y=3x^2+6x\)

Solution:

The Parabola given parameters are: \(a=3, b=6, c=0\)

Substitute the values in vertex formula: \((\frac{-b}{2a} , \frac{4ac-b^2}{4a})=(\frac{-6}{2(3)} , \frac{4(3)(0)-6^2}{4(3)})\)

Therefore, the vertex of the parabola is \((-1, 3)\).

To find the focus of the parabola, substitute the values in the focus formula: \((\frac{-b}{2a}, \frac{4ac-b^2+1}{4a})=(\frac{-6}{2(3)}, \frac{4(3)(0)-6^2+1}{4(3)})\)

Focus of parabola is \((-1, \frac{-35}{12})\).

Exercises for Finding the Focus, Vertex, and Directrix of Parabola

Find the vertex and focus of each parabola.

• \(\color{blue}{(y-2)^2=3(x-5)^2}\)
• \(\color{blue}{y=4x^2+x-1}\)
• \(\color{blue}{y=x^2+2x+3}\)
• \(\color{blue}{x=y^2-4y}\)

• \(\color{blue}{Vertex: (5, 2),}\) \(\color{blue}{focus: (5, \frac{25}{12})}\)
• \(\color{blue}{Vertex: (\frac{-1}{8}, \frac{-17}{16}), focus: (\frac{-1}{8}, -1)}\)
• \(\color{blue}{Vertex: (-1, 2), focus: (-1, \frac{9}{4})}\)
• \(\color{blue}{Vertex: (-4, 2), focus: (\frac{-15}{4}, 2)}\)