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.
Related Topic
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:
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.
- \(\color{blue}{(y-2)^2=3(x-5)^2}\)
- \(\color{blue}{y=4x^2+x-1}\)
- \(\color{blue}{y=x^2+2x+3}\)
- \(\color{blue}{x=y^2-4y}\)

- \(\color{blue}{Vertex: (5, 2),}\) \(\color{blue}{focus: (5, \frac{25}{12})}\)
- \(\color{blue}{Vertex: (\frac{-1}{8}, \frac{-17}{16}), focus: (\frac{-1}{8}, -1)}\)
- \(\color{blue}{Vertex: (-1, 2), focus: (-1, \frac{9}{4})}\)
- \(\color{blue}{Vertex: (-4, 2), focus: (\frac{-15}{4}, 2)}\)
More math articles
- How long is the SAT Test?
- How to Solve Powers of Products and Quotients? (+FREE Worksheet!)
- How to Use Algebra Tiles to Identify Equivalent Expressions?
- How to Evaluate Logarithms? (+FREE Worksheet!)
- Top 10 3rd Grade Georgia Milestones Assessment System Math Practice Questions
- Volume of Cubes
- FREE CLEP College Math Practice Test
- FREE 7th Grade SBAC Math Practice Test
- How to Write an Exponential Function: Word Problems
- Number Properties Puzzle -Critical Thinking 3
What people say about "How to Find the Focus, Vertex, and Directrix of a Parabola?"?
No one replied yet.