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.
Find the Focus, Vertex, and Directrix of a Parabola: what to notice and how to work it
What to notice first
Common student mistake
Key formulas and cues
A reliable path
- Match the formIdentify the conic by its equation pattern.
- Read featuresFind the center, vertex, radius, axes, foci, or asymptotes.
- Sketch from anchorsPlot key points first, then draw the curve.
Worked examples
Circle center and radius
- Compare to circle standard form.
- The center is (4, -1).
- The radius is the square root of 25.
Parabola direction
- The x part is squared.
- The parabola opens up or down.
- The positive coefficient means it opens up.
Try one before moving on
Find the Focus, Vertex, and Directrix of a Parabola: pop-up practice
“A parabola consists of three parts: the vertex, focus, and directrix. The vertex represents the point where the curve reaches its maximum or minimum value, while the focus is a fixed point located inside the parabola. The directrix lies outside the curve and runs parallel to its axis. If these geometric concepts seem tricky at first, you can to get step-by-step guidance and interactive explanations that make the process of learning much easier.
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.
- \(\color{blue}{(y-2)^2=3(x-5)^2}\)
- \(\color{blue}{y=4x^2+x-1}\)
- \(\color{blue}{y=x^2+2x+3}\)
- \(\color{blue}{x=y^2-4y}\)
- \(\color{blue}{Vertex: (5, 2),}\) \(\color{blue}{focus: (5, \frac{25}{12})}\)
- \(\color{blue}{Vertex: (\frac{-1}{8}, \frac{-17}{16}), focus: (\frac{-1}{8}, -1)}\)
- \(\color{blue}{Vertex: (-1, 2), focus: (-1, \frac{9}{4})}\)
- \(\color{blue}{Vertex: (-4, 2), focus: (\frac{-15}{4}, 2)}\)
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