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

You can easily find the focus, vertex, and directrix from the standard form of a parabola.

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.

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

  • The standard form of Parabola when it opens up or down is \((x- h)^2= 4p(y-k)\), where the focus is \(h,k+p\) and the directrix is \(y=k-p\).
  • The standard form of Parabola when it opens right or left is \((y+k)^2= 4p(x-h)\), where the focus is \(h+p,k\) and the directrix is \(x=h-p\).
    • 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:

    Find 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 parabola is \((-1, 3)\).

    To find focus of parabola, substitute the values in 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 vertex and focus of each parabola.

    • \(\color{blue}{(y-2)^2=3(x-5)^2}\)
    • \(\color{blue}{y=4x^2+x-1}\)
    • \(\color{blue}{y=x^2+2x+3}\)
    • \(\color{blue}{x=y^2-4y}\)
    This image has an empty alt attribute; its file name is answers.png
    • \(\color{blue}{Vertex: (5, 2),}\) \(\color{blue}{focus: (5, \frac{25}{12})}\)
    • \(\color{blue}{Vertex: (\frac{-1}{8}, \frac{-17}{16}), focus: (\frac{-1}{8}, -1)}\)
    • \(\color{blue}{Vertex: (-1, 2), focus: (-1, \frac{9}{4})}\)
    • \(\color{blue}{Vertex: (-4, 2), focus: (\frac{-15}{4}, 2)}\)

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