How to find vertex with focus and directrix

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

• 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:

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.

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

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

A parabola is set of all points in a plane which are an equal distance away from a given point and given line. The point is called the focus of the parabola and the line is called the directrix.

The focus lies on the axis of symmetry of the parabola.

Finding the focus of a parabola given its equation

If you have the equation of a parabola in vertex form y=a(x−h )2+k, then the vertex is at (h,k) and the focus is (h,k+14a).

Notice that here we are working with a parabola with a vertical axis of symmetry, so the x-coordinate of the focus is the same as the x-coordinate of the vertex.

Example 1:

Find the focus of the parabola y=18x2.

Here h=0 and k=0, so the vertex is at the origin. The coordinates of the focus are (h,k+ 14a) or (0,0+14a).

Since a=18, we have

14a=1(12 )

=2

The focus is at (0,2).

Example 2:

Find the focus of the parabola y=−( x−3)2−2.

Here h=3 and k=−2, so the vertex is at (3,−2). The coordinates of the focus are (h,k+14a ) or (3,−2+14a).

Here a=−1, so

−2+14a=−2−14

=−2.25

The focus is at (3,−2.25).

Given the focus and directrix of a parabola , how do we find the equation of the parabola?

If we consider only parabolas that open upwards or downwards, then the directrix will be a horizontal line of the form y = c .

Let ( a , b ) be the focus and let y = c be the directrix. Let ( x 0 , y 0 ) be any point on the parabola.

Any point, ( x 0 , y 0 ) on the parabola satisfies the definition of parabola, so there are two distances to calculate:

1. Distance between the point on the parabola to the focus
2. Distance between the point on the parabola to the directrix

To find the equation of the parabola, equate these two expressions and solve for y 0 .

Find the equation of the parabola in the example above.

Distance between the point ( x 0 , y 0 ) and ( a , b ) :

( x 0 − a ) 2 + ( y 0 − b ) 2

Distance between point ( x 0 , y 0 ) and the line y = c :

| y 0 − c |

(Here, the distance between the point and horizontal line is difference of their y -coordinates.)

Equate the two expressions.

( x 0 − a ) 2 + ( y 0 − b ) 2 = | y 0 − c |

Square both sides.

( x 0 − a ) 2 + ( y 0 − b ) 2 = ( y 0 − c ) 2

Expand the expression in y 0 on both sides and simplify.

( x 0 − a ) 2 + b 2 − c 2 = 2 ( b − c ) y 0

This equation in ( x 0 , y 0 ) is true for all other values on the parabola and hence we can rewrite with ( x , y ) .

Therefore, the equation of the parabola with focus ( a , b ) and directrix y = c is

( x − a ) 2 + b 2 − c 2 = 2 ( b − c ) y

Example:

If the focus of a parabola is ( 2 , 5 ) and the directrix is y = 3 , find the equation of the parabola.

Let ( x 0 , y 0 ) be any point on the parabola. Find the distance between ( x 0 , y 0 ) and the focus. Then find the distance between ( x 0 , y 0 ) and directrix. Equate these two distance equations and the simplified equation in x 0 and y 0 is equation of the parabola.

The distance between ( x 0 , y 0 ) and ( 2 , 5 ) is ( x 0 − 2 ) 2 + ( y 0 − 5 ) 2

The distance between ( x 0 , y 0 ) and the directrix, y = 3 is

| y 0 − 3 | .

Equate the two distance expressions and square on both sides.

( x 0 − 2 ) 2 + ( y 0 − 5 ) 2 = | y 0 − 3 |

( x 0 − 2 ) 2 + ( y 0 − 5 ) 2 = ( y 0 − 3 ) 2

Simplify and bring all terms to one side:

x 0 2 − 4 x 0 − 4 y 0 + 20 = 0

Write the equation with y 0 on one side:

y 0 = x 0 2 4 − x 0 + 5

This equation in ( x 0 , y 0 ) is true for all other values on the parabola and hence we can rewrite with ( x , y ) .

So, the equation of the parabola with focus ( 2 , 5 ) and directrix is y = 3 is

y = x 2 4 − x + 5

How do you find the vertex when given focus?

Is the vertex in between the focus and Directrix?