How to Understand the Equation of a Parabola, Directrix and Focus

Updated on October 14, 2019
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Eugene is a qualified control/instrumentation engineer Bsc (Eng) and has worked as a developer of electronics & software for SCADA systems.

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Definition of a Parabola

"A locus is a curve or other figure formed by all the points satisfying a particular equation."

One way we can define a parabola is that it is the locus of points that are equidistant from both a line called the directrix and a point called the focus. So each point is the same distance from the focus as it is from the directrix as you can see in the animation below.

A parabola is a locus of points equidistant (the same distance) from a line called the directrix and point called the focus.
A parabola is a locus of points equidistant (the same distance) from a line called the directrix and point called the focus. | Source

Definition of a Parabola

A parabola is a locus of points equidistant from a line called the directrix and point called the focus.

A Parabola is a Conic Section

Another way of defining a parabola

When a plane intersects a cone, we get different shapes or conic sections where the plane intersects the outer surface of the cone. If the plane is parallel to the bottom of the cone, we just get a circle. As the angle A in the animation below changes, it eventually becomes equal to B and the conic section is a parabola.

A parabola is the shape produced when a plane intersects a cone and the angle of intersection to the axis is equal to half the opening angle.
A parabola is the shape produced when a plane intersects a cone and the angle of intersection to the axis is equal to half the opening angle. | Source
Conic sections.
Conic sections. | Source

Equations of Parabolas

There are several ways we can express the equation of a parabola:

  • As a quadratic function
  • Vertex form
  • Focus form

We'll explore these later, but first let's look at the simplest parabola.

The Simplest Parabola y = x²

The simplest parabola with the vertex at the origin, point (0,0) on the graph, has the equation y = x².

The value of y is simply the value of x multiplied by itself.

x
y = x²
1
1
2
4
3
9
4
16
5
25

Graph of y = x² - The Simplest Parabola

The simplest parabola, y = x²
The simplest parabola, y = x² | Source

Let's Give x a Coefficient!

The simplest parabola is y = x2 but if we give x a coefficient, we can generate an infinite number of parabolas with different "widths" depending on the value of the coefficient ɑ.

So lets make y = ɑx2

In the graph below, ɑ has various values. Notice that when ɑ is negative, the parabola is "upside down". We'll discover more about this later. Remember the y = ɑx2 form of the equation of a parabola is when its vertex is at the origin.

Making ɑ smaller results in a "wider" parabola. If we make ɑ bigger, the parabola gets narrower.

Parabolas with different coefficients of x²
Parabolas with different coefficients of x² | Source

Turning the Simplest Parabola on Its Side

If we turn the parabola y = x2 on its side, we get a new function y2 = x or x = y2. This just means we can think of y as being the independent variable and squaring it gives us the corresponding value for x.

So:

When y = 2, x = y2 = 4

when y = 3, x = y2 = 9

when y = 4, x = y2 = 16

and so on...

The parabola x = y²
The parabola x = y² | Source

Just like the case of the vertical parabola, we can again add a coefficient to y2.

So we have x = ɑy2

Parabolas with different coefficients of y²
Parabolas with different coefficients of y² | Source

Vertex Form of a Parabola Parallel to Y Axis

One way we can express the equation of a parabola is in terms of the coordinates of the vertex. The equation depends on whether the axis of the parabola is parallel to the x or y axis, but in both cases, the vertex is located at the coordinates (h,k). In the equations, ɑ is a coefficient and can have any value.

When the axis is parallel to y axis:

y = ɑ(x - h)2 + k

if ɑ = 1 and (h,k) is the origin (0,0) we get the simple parabola we saw at the start of the tutorial:

y = 1(x - 0)2 + 0 = x2

Vertex form of the equation of a parabola.
Vertex form of the equation of a parabola. | Source

When the axis is parallel to the x axis:

x = ɑ(y - h)2 + k

Notice that this doesn't give us any information about the location of the focus or directrix.

Vertex form of the equation of a parabola.
Vertex form of the equation of a parabola. | Source

Equation of a Parabola in Terms of the Coordinates of the Focus

Another way of expressing the equation of a parabola is in terms of the coordinates of the vertex (h,k) and the focus.

We saw that:

y = ɑ(x - h)2 + k

Using Pythagoras's Theorem we can prove that the coefficient ɑ = 1/4p, where p is the distance from the focus to the vertex. Substituting this into the equation above gives us:

When the axis of symmetry is parallel to y axis:

y = ɑ(x - h)2 + k = 1/(4p)(x - h)2 + k

Multiply both sides of the equation by 4p:

4py = (x - h)2 + 4pk

Rearrange:

4p(y - k) = (x - h)2

or

(x - h)2 = 4p(y - k)

Similarly:

When the axis of symmetry is parallel to x axis:

(y - k)2 = 4p(x - h)


Equation of a parabola in terms of the focus.
Equation of a parabola in terms of the focus. | Source
Focus form of the equation of a parabola.
Focus form of the equation of a parabola. | Source

A Quadratic Function is a Parabola

Consider the function y = ɑx2 + bx + c

This is called a quadratic function because of the square on the x variable.

This is another way we can express the equation of a parabola.


How to Determine Which Direction a Parabola Opens

Irrespective of which form of equation that is used to describe a parabola, the coefficient of x2 determines whether a parabola will "open up" or "open down". Open up means that the parabola will have a minimum and the value of y will increase on both sides of the minimum. Open down means it will have a maximum and the value of y decreases on both sides of the max.

  • If ɑ is positive, the parabola will open up
  • If ɑ is negative the parabola will open down

Parabola Opens Up or Opens Down

The sign of the coefficient of x² determines whether a parabola opens up or opens down.
The sign of the coefficient of x² determines whether a parabola opens up or opens down. | Source

How to Find the Vertex of a Parabola

  1. Expand out the equation into the form ɑx2 + bx + c
  2. Identify the coefficients a and b
  3. The vertex occurs at x = -b/2ɑ
  4. Plug the value -b/2ɑ into the equation to get the value of y

Example: Find the vertex of the equation y = 5x2 - 10x + 7

  1. The coefficient a is positive, so the parabola opens up and the vertex is a minimum
  2. ɑ = 5 and b = -10 so the minimum occurs at -b/2ɑ = - (-10)/(2(5)) = 1
  3. Substitute for x giving:

y = 5x2 - 10x + 7

= 5(1)2 - 10(1) + 7

= 5 - 10 + 7

= 2

So the vertex occurs at (1,2)

How to Find the X-Intercepts of a Parabola

A quadratic function y = ɑx2 + bx + c is the equation of a parabola.

If we set the quadratic function to zero, we get a quadratic equation

i.e. ɑx2 + bx + c = 0 .

Graphically, equating the function to zero means setting a condition of the function such that the y value is 0, in other words, where the parabola intercepts the x axis.

The solutions of the quadratic equation allow us to find these two points. If there are no real number solutions, i.e. the solutions are imaginary numbers, the parabola doesn't intersect the x axis.

The solutions of a quadratic equation are given by the equation:

x = -b ± √(b2 -4ac) / 2ɑ


The roots of a quadratic equation give the x axis intercepts for a parabola.
The roots of a quadratic equation give the x axis intercepts for a parabola. | Source
A and B are the x-intercepts of the parabola y = ax² + bx + c and roots of the quadratic equation ax² + bx + c = 0
A and B are the x-intercepts of the parabola y = ax² + bx + c and roots of the quadratic equation ax² + bx + c = 0 | Source

Example 1: Find the x-axis intercepts of the parabola y = 3x2 + 7x + 2

Solution

  • y = ɑx2 + bx + c
  • In our example y = 3x2 + 7x + 2
  • Identify the coefficients and constant c
  • So ɑ = 3, b = 7 and c = 2
  • The roots of the quadratic equation 3x2 + 7x + 2 = 0 are at x = -b ± √(b2 - 4ɑc) / 2ɑ
  • Substitute for ɑ, b and c
  • The first root is at x = -7 + √(72 -4(3)(2)) / (2(3) = -1/3
  • The second root is at -7 - √(72 -4(3)(2)) / (2(3) = -2
  • So the x axis intercepts occur at (-2, 0) and (-1/3, 0)


Example 1: Find the x-intercepts of the parabola y = 3x2 + 7x + 2
Example 1: Find the x-intercepts of the parabola y = 3x2 + 7x + 2 | Source

Example 2: Find the x-axis intercepts of the parabola with vertex located at (4, 6) and focus at (4, 3)

Solution

  • The equation of the parabola in focus vertex form is (x - h)2 = 4p(y - k)
  • The vertex is at (h,k) giving us h = 4, k = 6
  • The focus is located at (h, k + p). In this example the focus is at (4, 3) so k + p = 3. But k = 6 so p = 3 - 6 = -3
  • Plug the values into the equation (x - h)2 = 4p(y - k) so (x - 4)2 = 4(-3)(y - 6)
  • Simplify giving (x - 4)2 = -12(y - 6)
  • Expand out the equation gives us x2 - 8x + 16 = -12y + 72
  • Rearrange 12y = -x2 + 8x + 56
  • Giving y = -1/12x2 + 2/3x + 14/3
  • The coefficients are a = -1/12, b = 2/3, c = 14/3
  • The roots are at -2/3 ± √((2/3)2 - 4(-1/12)(14/3))/(2(-1/12)
  • This gives us x = -4.49 approx and x = 12.49 approx
  • So the x axis intercepts occur at (-4.49, 0) and (12.49, 0)

Example 2: Find the x-intercepts of the parabola with vertex at (4, 6) and focus at (4, 3)
Example 2: Find the x-intercepts of the parabola with vertex at (4, 6) and focus at (4, 3) | Source

How to Find the Y-Intercepts of a Parabola

To find the y-axis intercept (y-intercept) of a parabola, we set x to 0 and calculate the value of y.

A is the y-intercept of the parabola y = ax² + bx + c
A is the y-intercept of the parabola y = ax² + bx + c | Source

Example 3: Find the y-intercept of the parabola y = 6x2 + 4x + 7

Solution:

y = 6x2 + 4x + 7

Set x to 0 giving

6(0)2 + 4(0) + 7 = 7

The intercept occurs at (0, 7)

Example 3: Find the y-intercept of the parabola y = 6x² + 4x + 7
Example 3: Find the y-intercept of the parabola y = 6x² + 4x + 7 | Source

Summary of Parabola Equations

Equation Type
Axis Parallel to Y-Axis
Axis Parallel to X-Axis
Quadratic Function
y = ɑx² + bx + c
x = ɑy² + by + c
Vertex Form
y = ɑ(x - h)² + k
x = ɑ(y - h)² + k
Focus Form
(x - h)² = 4p(y - k)
(y - k)² = 4p(x - h)
Parabola with Vertex at the Origin
y² = 4px
x² = 4py

How the Parabola is Used in the Real World

The parabola isn't just confined to math. The parabola shape appears in nature and we use it in science and technology because of its properties.

  • When you kick a ball into the air or a projectile is fired, the trajectory is a parabola
  • The reflectors of vehicle headlights or flashlights are parabolic shaped
  • The mirror in a reflecting telescope is parabolic
  • Satellite dishes are in the shape of a parabola as are radar dishes

Related reading: Deriving Projectile Motion Equations

Radar dishes and radio telescopes are parabolic shaped.
Radar dishes and radio telescopes are parabolic shaped. | Source
Water from a fountain (which can be considered as a stream of particles) follows a parabolic trajectory
Water from a fountain (which can be considered as a stream of particles) follows a parabolic trajectory | Source

Acknowledgements

All graphics were created using GeoGebra Classic.

© 2019 Eugene Brennan

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    • megalos profile image

      rafia 

      2 weeks ago from lahore pakistna

      hey you are a nice teacher! i previously did not know the open up or down graph ..but now i do ..thank you!

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