Parabola

A parabola is defined as a collection of points such that the distance to a fixed point (the focus) and a fixed straight line (the directrix) are equal. But it's probably easier to remember it as the U-shaped curved line created when a quadratic is graphed.

Many real-world objects travel in a parabolic shape. When you shoot a basketball, the path of the ball creates a parabola.

Example

A parabola that is said to open upwards is shaped like a "U," while a parabola said to open downwards is shaped like an upside-down U.

Example

Opens upwards
Opens downwards

A parabola does not have to be vertical, but horizontal parabolas are not functions (they fail the vertical line test). Some functions you see will be horizontal parabolas with restricted domains (). You will learn more about these when you explore conics.

Example

x = -y2
x = y2

The vertex of a parabola is the point where the parabola changes direction, and where the graph is most curved. On graphs of quadratics, it is found at the very top or bottom of the quadratic. The vertex is the point of the parabola at the axis of symmetry.

For quadratics in the standard form ax2 + bx + c, the axis of symmetry can be found using the equation x = . To find the y-coordinate of the vertex, find the axis of symmetry and substitute that x-value into the original equation.

Example

f(x) = x2 + 6x + 11
a = 1; b = 6; c = 11
x = = -3
f(-3) = (-3)2 + 6(-3) + 11
f(-3) = 9 - 18 + 11 = 2
Vertex at (-3,2)

calculating the focus and directrix


Below is an example of how to calculate the focus and directrix that may provide a better understanding of the mathematical definition of a parabola provided above:

Example

The focus is a point located on the same line as the axis of symmetry, while the directrix is a line perpendicular to the axis of symmetry. For parabolas, the focus is always on the inside of the parabola, and the directrix never touches the parabola. Because the vertex is the same distance from the focus and directrix, the directrix has a location directly opposite of the focus.

For a parabola in the vertex form y = a(x - h)2 + k, the focus is located at (h, k + ) and the directrix is located at y = k - .

Example

y = (x - 3)2 + 2
h = 3 ; k = 2 ; a =
Focus: (3,2 + ) = (3,4)
Directrix: y = 2 - = 0

For horizontal parabolas, the vertex is x = a(y - k)2 + h, where (h,k) is the vertex. The focus of parabolas in this form have a focus located at (h + , k) and a directrix at x = h - . The axis of symmetry is located at y = k.


vertex form of a parabola


The vertex form of a parabola is another form of the quadratic function f(x) = ax2 + bx + c. The vertex form of a parabola is:

f(x) = a(x - h)2 + k

The a in the vertex form of a parabola corresponds to the a in standard form. If a is positive, the parabola will open upwards. If a is negative, the parabola will open downwards.

In vertex form, (h,k) describes the vertex of the parabola and the parabola has a line of symmetry x = h.

Vertex form is very similar to the general expression for function transformations. Vertex form makes it much easier to graph a parabola because it makes it easy to plot the vertex.

Example

f(x) = -(x - 1)2 + 4

From this equation, we can already tell that the vertex of the parabola is at (1,4), and the axis of symmetry is at x = 1. Now all that has to be done is to plug in points around the vertex, then graph.

You can use completing the square to convert a quadratic in standard form into vertex form. You can also convert to vertex form by using the knowledge that the vertex lies on the axis of symmetry.

The x-value of the vertex is x = - , so the y-value is y = f (-). Plug these values in for h and k in the vertex form equation. Then, a is the same in both forms, so simply copy that to the vertex form equation.

Example

f(x) = x2 - x - 20

a = 1; b = -1; c = -20

x = h = - = - =

f() = ()2 - - 20 = - - 20 = -20

f(x) = (x - )2 - 20

To convert a parabola in vertex form to standard form, expand the equation and simplify.

Example

f(x) = 5(x + 3)2 - 6

f(x) = 5(x2 + 6x + 9) - 6

f(x) = 5x2 + 30x + 45 - 6

f(x) = 5x2 + 30x + 39


See also quadratic function, function transformations, completing the square.