# Equation of Parabola

## Definition and Equation of a Parabola with Vertical Axis

A parabola is the set of all points \( M(x,y)\) in a plane such that the distance from \( M \) to a fixed point \( F \) called the focus is equal to the distance from \( M \) to a fixed line called the directrix as shown below in the graph.

Let us consider a parabola with a vertex \( V(0,0) \) (the lowest point) at the origin (0,0) as shown in the graph and the focus \( F(0 , p) \) on the axis of symmetry (the y axis) with \( p > 0 \).

The distance between the points \(M(x,y) \) on the parabola and the focus \( F(0 , p)\) is given by

\( MF = \sqrt{(x -0)^2 + (y - p)^2} \)

The distance from point \(M(x,y) \) to the directix of equation \( y = - p \) is given by

\( MD = y + p \)

According to the above definition of the parabola these two distances are equal; hence

\(\sqrt{(x -0)^2 + (y - p)^2} = y + p\)

Square both sides and expand the two sides of the equation

\( x^2 + y^2 - 2 py + p^2 = y^2 + 2 py + p^2 \)

Group like term

\( 4 py = x^2 \)

Write the equation of the parabola as \( y \) in terms of \( x \).

\( y = \dfrac{1}{4p} x^2 \)

Example 1

Point \( ( 4,2) \) is on the graph of a parabola with vertex at the origin \( (0,0) \) and vertical axis. Find the focus of the parabola, graph it and label the focus and graph the directrix.

__Solution to Example 1__

The equation of a parabola with vertical axis at whose vertex is at the origin is given by

\( y = \dfrac{1}{4p} x^2 \)

Since \( ( 4,2) \) is on the graph of the parabola, the coordinates \( x = 4 \) and \( y = 2 \) satisfy the equation of the parabola. Hence

\( 2 = \dfrac{1}{4p} (4)^2 \)

Simplify

\( 2 = \dfrac{16}{4p} \)

Solve for \( p \)

\( p = 2 \)

The focus is at the point \( F(0 , 2)\) and the directrix is given by the horizontal line \( y = - 2 \) as shown in the graph below.

We can generalize and write the equation of a parabola at a vertex \( V(h,k) \) as follows

\( y = \dfrac{1}{4p} (x - h)^2 + k\)

with vertex \( V(h,k) \) and focus \( F(h,k+p) \) and directrix given by the equation \( y = k - p \)Example 2

Find the vertex, focus and directrix of the parabola given by the equation \(y = \dfrac{1}{16} x^2 - \dfrac{1}{4} x + \dfrac{9}{4}\) .

__Solution to Example 2__

Rewrite the given equation in standard form by completing the square. factor \( 1/16 \) out of the terms in \( x \) and \( x^2 \)

\(y = \dfrac{1}{16} (x^2 - 4 x) + \dfrac{9}{4}\) .

Complete the square inside the parentheses

\(y = \dfrac{1}{16} ((x-2)^2 - 2^2 ) + \dfrac{9}{4}\)

Rewrite in standard form

\(y = \dfrac{1}{16} ((x-2)^2 - 4 ) - \dfrac{1}{4} + \dfrac{9}{4}\)

Group like terms

\(y = \dfrac{1}{16} (x - 2)^2 + 2 \)

Compare the above equation to the standard form \( y = \dfrac{1}{4p} (x - h)^2 + k\) and identify the parameters \( p \), \( h \) and \( k \)

\( \dfrac{1}{16} = \dfrac{1}{4p}\); solve for \( p \) to obtain \( p = 4 \)

\( h = 2 \) and \( k = 2 \)

Vertex at \( V(h,k) = V(2,2)\), Focus at \( F(h,k+p) = F(2,6)\) , directrix given by \( y = k - p = - 2 \)

## Equation of a Parabola with Horizontal Axis

The equation of a parabola with a horizontal axis is written as

\( x = \dfrac{1}{4p} (y - k)^2 + h\)

with vertex \( V(h,k) \) and focus \( F(h+p,k) \) and directrix given by the equation \( x = h - p \)Example 3

Find the vertex, focus and directrix of the parabola given by the equation \(x = \dfrac{1}{4} y^2 - y + 11\) .

__Solution to Example 3__

Group the terms in \( y^2 \) and \(y \) and factor \( 1/4 \) out.

\(x = \dfrac{1}{4} (y^2 - 4 y) + 11\)

Use the terms \( y^2 \) and \(y \) inside the parentheses and complete the square

\(x = \dfrac{1}{4} ((y^2 - 2) - 2^2) + 11\)

Rewrite in standard form

\(y = \dfrac{1}{4} ((y-2)^2) + 10 \)

Group like terms

Compare the above equation to the equation in standard form \( x = \dfrac{1}{4p} (y - k)^2 + h\) and identify the parameters \( p \), \( h \) and \( k \)

\( \dfrac{1}{4p} = \dfrac{1}{4} \) gives \( p = 1 \)

\( h = 10 \) and \( k = 2 \)

Vertex at \( V(h,k) = V(10,2)\), Focus at \( F(h+p,k) = F(11,2)\) , directrix given by \( x = h - p = 9 \)

## Interactive Turorial on Equation of a Parabola

An app to explore the equation of a parabola and its properties is now presented. The equation used is the standard equation that has the form

\( y = \dfrac{1}{4 p}(x - h)^2 + k \)

where h and k are the x- and y-coordinates of the vertex of the parabola and p is a non zero real number.The exploration is carried out by changing the parameters \( p, h \) and \( k \) included in the above equation and carry out the activities described below.

The default values when you open this page are: \( p = 1, h = 2 \) and \( k = 3 \)

Click on the button "Plot Equation" to start.

Hover the mousse cursor on the graph or plotted point to read the coordinates.

1 - Start with the default values \( p = 1, h = 2 \) and \( k = 3 \) the button "Plot Equation". Hover the mousse cursor over the graph to trace and read the coordinates of points on the graph, on the focus F or vertex V.

a) Use the values of \( p = 1, h = 2 \) and \( k = 3 \) and calculate the coordinates of the focus \( F \), the vertex \( V \) and the equation of the directrix and compare them to the graphical values.

b) Select a point \( M \) on the parabola and find the distance \( MF \) and compare it to the the distance from \( M \) to the directrix.(see definition of parabola above). Are they equal?(or close)

2 - On paper, find the equation of the parabola for the values \( p = 4, h = 1 \) and \( k = - 4 \).

a) Calculate the coordinates of the focus \( F \), the vertex \( V \) and the equation of the directrix

b) Calculate the x and y intercepts

c) Set the values \( p = 4, h = 1 \) and \( k = - 4 \) in the app above and then read and check the equation of the parabola, the coordinates of the focus \( F \) and vertex \( V \) and the equation of the directrix.

d) Check the x and y intercepts

3 -

__Exercise:__

a) On paper, rewrite the equation

\[ x^2 - 4 x - 4 y = 0 \]

in the form \( y = \dfrac{1}{4 p} (x - h)^2 + k \) (see example 2 above)

b) Identify and find the values of \( p \), \( h \) and \( k \).

c) Find the coordinates of the focus \( F \), the vertex \( V \), the x and y intercepts and the equation of the directrix

d) Use the app above and check the values found by calculations.

If needed, Free graph paper is available.

## More References and Links to Topics Related to the Equation of the Parabola

Tutorial on How Parabolic Dish Antennas work?Tutorial on how to Find The Focus of Parabolic Dish Antennas .

Use of parabolic shapes as Parabolic Reflectors and Antennas .

Interactive tutorial on how to find the equation of a parabola .

Define and Construct a Parabola .

Three Points Parabola Calculator.

Similar tutorials on circle ,

Ellipse

and the hyperbola can be found in this site.