How do you decide how wide the parabola is?

Question

How do you decide how wide the parabola is?

Answer 1

Width of a parabola isn't typically one of the things you look for when finding its properties. The most significant properties of a parabola are:

(1) x-intercepts, and
(2) vertex

For a parabola y = A(x-h)^2 + k,
the constant that determines the width (and whether it's facing up or down) is A. For larger values of A, the parabola will be narrow. For small values of A, the parabola will be wide. For negative values of A, the parabola will face downward, and for positive values of A, the parabola will face upward.

Answer 2

The "width" of the parabola is usually measured across the focus as it is clearly variable.
This special chord is called the 'latus rectum' and its length is equal to 2A where A = focal length of the parabola.

(ie the distance from the vertex to the focus along the axis)

A parabola with a vertical axis and vertex at (h, k) can be expressed in the following form:

(x - h)² = 4A(y - k)

If the parabola has a horizontal axis and the same vertex then it is expressed as:

(y - k)² = 4A(x - h)

As A increases the "width" of the parabola increases

The size of the latus rectum is thus inversely related to the coefficient of x² (or y²) in the general quadratic form

Answer 3

if you put it in the format

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

or

y=ax^2+dx+e

a will tell you how "wide" the parabola is. a>1 makes it skinny, a<1 makes it fat. Wide depends on where you measure the width.

Answer 4

use the focal diameter
the focal diameter is 4p and is measured by crossing at the focus perpendicular to either the x or y axis