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 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.
- \(\color{blue}{(y-2)^2=3(x-5)^2}\)
- \(\color{blue}{y=4x^2+x-1}\)
- \(\color{blue}{y=x^2+2x+3}\)
- \(\color{blue}{x=y^2-4y}\)
- \(\color{blue}{Vertex: (5, 2),}\) \(\color{blue}{focus: (5, \frac{25}{12})}\)
- \(\color{blue}{Vertex: (\frac{-1}{8}, \frac{-17}{16}), focus: (\frac{-1}{8}, -1)}\)
- \(\color{blue}{Vertex: (-1, 2), focus: (-1, \frac{9}{4})}\)
- \(\color{blue}{Vertex: (-4, 2), focus: (\frac{-15}{4}, 2)}\)
Related to This Article
More math articles
- 10 Most Common 5th Grade NYSE Math Questions
- How to Write a Formula for a Recursive Sequence
- A Complete Explanation of the Continuity over an Interval
- PSAT Cаlсulаtоr Pоliсу
- Top 10 TSI Math Test-taking Tips
- 5 Best CHSPE Math Study Guides
- Top 10 Math Books for 8th Graders: A Roadmap to Excellence
- Journey to the Edge: A Meticulous Exploration of One-Sided Limits
- How long Is the CBEST Test Valid?
- What Kind of Math Courses Are Required for Business Degree?
What people say about "How to Find the Focus, Vertex, and Directrix of a Parabola? - Effortless Math: We Help Students Learn to LOVE Mathematics"?
No one replied yet.