# 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.

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.

## 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.

• $$\color{blue}{(y-2)^2=3(x-5)^2}$$
• $$\color{blue}{y=4x^2+x-1}$$
• $$\color{blue}{y=x^2+2x+3}$$
• $$\color{blue}{x=y^2-4y}$$
• $$\color{blue}{Vertex: (5, 2),}$$ $$\color{blue}{focus: (5, \frac{25}{12})}$$
• $$\color{blue}{Vertex: (\frac{-1}{8}, \frac{-17}{16}), focus: (\frac{-1}{8}, -1)}$$
• $$\color{blue}{Vertex: (-1, 2), focus: (-1, \frac{9}{4})}$$
• $$\color{blue}{Vertex: (-4, 2), focus: (\frac{-15}{4}, 2)}$$

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