How to Find the Focus, Vertex, and Directrix of a Parabola?

“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

• How to Write the Equation of Parabola

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

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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.

• \(\color{blue}{(y-2)^2=3(x-5)^2}\)
• \(\color{blue}{y=4x^2+x-1}\)
• \(\color{blue}{y=x^2+2x+3}\)
• \(\color{blue}{x=y^2-4y}\)

• \(\color{blue}{Vertex: (5, 2),}\) \(\color{blue}{focus: (5, \frac{25}{12})}\)
• \(\color{blue}{Vertex: (\frac{-1}{8}, \frac{-17}{16}), focus: (\frac{-1}{8}, -1)}\)
• \(\color{blue}{Vertex: (-1, 2), focus: (-1, \frac{9}{4})}\)
• \(\color{blue}{Vertex: (-4, 2), focus: (\frac{-15}{4}, 2)}\)