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?
Tutor-style math help

Find the Focus, Vertex, and Directrix of a Parabola: what to notice and how to work it

Conics skill
Conic sections are graph shapes with standard forms. The equation tells you the center or vertex, then the key distances shape the graph.

What to notice first

Name the conic first. Circle, ellipse, parabola, and hyperbola have different standard forms and different graph features.

Common student mistake

Do not read signs backward in \((x-h)\) and \((y-k)\). The center or vertex uses \(h\) and \(k\), not the visible sign alone.

Key formulas and cues

\((x-h)^2+(y-k)^2=r^2\)
\(\frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1\)
\((y-k)=a(x-h)^2\)
\(\frac{(x-h)^2}{a^2}-\frac{(y-k)^2}{b^2}=1\)
vertex axis

A reliable path

  1. Match the formIdentify the conic by its equation pattern.
  2. Read featuresFind the center, vertex, radius, axes, foci, or asymptotes.
  3. Sketch from anchorsPlot key points first, then draw the curve.

Worked examples

Circle center and radius

Example: \((x-4)^2+(y+1)^2=25\)
  1. Compare to circle standard form.
  2. The center is (4, -1).
  3. The radius is the square root of 25.
Answer: Center (4, -1), radius 5

Parabola direction

Example: \((x-2)^2=8(y+3)\)
  1. The x part is squared.
  2. The parabola opens up or down.
  3. The positive coefficient means it opens up.
Answer: Opens up
Open pop-up practice
Try one before moving on
Try: Find the center of \((x+3)^2+(y-2)^2=16\).
Answer: (-3, 2).
Next step: do the matching worksheet or quiz while the method is still fresh, then come back and explain the first step in your own words.

“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

  • 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 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: \((\frac{-b}{2a}, \frac{4ac-b^2}{4a})=(\frac{-6}{2(3)}, \frac{4(3)(0)-6^2}{4(3)})\)

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    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.

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

    Effortless Math Team

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