A subclass of polynomial function where the variable x is in the second power (i.e. squared) (i.e. a degree two polynomial in one variable (i.e. univariate)). a, b and c are any real numbers, a≠0 (b and/or c can be 0).
Its graph in the XY Cartesian plane is a parabola, a parabolic curve,
sort of a smooth U, bowl, wok, (golden) arch, shield shape.
Parabola is one of the conic sections (slice thru a cone),
along with circles, ellipses, hyperbolas.
Example with all the points of interest:
The lowest or highest point on the parabola,
i.e. the smallest / minimum / nadir or largest / maximum / zenith value of the function,
is the vertex.
a>0: the graph is opening upward. concave up. Vertex is the minimum.
a<0: the graph is opening downward. concave down. Vertex is the maximum.
The x-coordinate of the vertex is the average of the two x-intercepts,
i.e. midway between them, which is -b/2a.
The y-coordinate of the vertex is the function evaluated at this x:
ƒ(-b/2a) = 4ac-b^{2}/4a = c-b^{2}/4a
So the vertex is (-b/2a,ƒ(-b/2a))
Parabola is symmetric about the vertical line [x=-b/2a] through its vertex.
Parabola crosses the y axis at its y-intercept: (0,c).
Every parabola has one y-intercept.
Parabola might cross the X axis at the x-intercept(s),
also called the zeroes or the roots of the function.
X-intercepts' x coordinate(s) is/are the solution(s), if any,
to the quadratic equation ax^{2} + bx + c = 0. (i.e. y is 0)
There can be 0, 1 or 2 real number solutions:
b^{2}-4ac is the discriminant which indicates how many (real number)
zeroes/roots there are:
2 if the discriminant b^{2}-4ac > 0. (b>√(ac))
x=(-b ± √(b^{2}-4ac)) / 2a
[quadratic formula]
1 if the discriminant b^{2}-4ac = 0. (b=√(ac))
x=-b/2a.
Parabola's vertex is on the X axis.
(the quadratic equation can be written a(x+b/2a)^{2}=0)
0 if the discriminant b^{2}-4ac < 0. Parabola is completely above or below
the X axis. (The quadratic equation has two imaginary/complex number solutions as
complex conjugates p±qi.)
The sum of the solutions is -b/a.
The product of the solutions is c/a.
If the parabola crosses the X axis ,i.e.
the quadratic equation has 2 real roots, then
both roots can be positive, both negative, or one positive and one negative:
The focus of the parabola is a point on the axis of symmetry
(i.e. same x as the vertex) that is 1/(4a) away: (-b/2a,(4ac-b^{2}+1)/4a)
It's in the concave side.
If the parabola can reflect light/radio/sound/particles, then that stuff which travels parallel
to the axis of symmetry and strikes its concave side is reflected to the focus.
Conversely, that stuff that originates at the focus is reflected
into a parallel beam, i.e. it can be "aimed" (gain).
This property is used in reflecting telescopes, searchlights, flashlights,
headlamps, solar cookers, satellite dishes, parabolic antennas.
A parabola rotated/spun 360° forms a 3D surface called a paraboloid.
A geometric definition of parabola as a conic section.
Effect of varying the constant c:
Area of the unique parabola that fits in a rectangle:
roots | Y intercept | vertex | area | distance between roots | mean value | derivative | integral ∫ | |
---|---|---|---|---|---|---|---|---|
±ax^{2}∓c | √(c/a) and -√(c/a) | c | (0,c) | 4/3 c √(c/a) | 2√(c/a) | 2/3 c | 2ax | ax^{3}/ 3 + cx |
ax^{2}+bx | 0 and -b/a | 0 | (-b/ 2a, -b^{2}/ 4a) | b^{3}/ 6a^{2} | b/a | b^{2}/ 6a | 2ax+b | ax^{3}/ 3 + bx^{2}/ 2 |
ax^{2}+bx+c,
b^{2}>4ac | (-b ± √(b^{2}-4ac)) / 2a | c | (-b/ 2a, c- b^{2}/ 4a) | √(b^{2}-4ac)(4ac-b^{2}) / 6a^{2} | √(b^{2}-4ac) / a | 4ac-b^{2} / 6a | 2ax+b | ax^{3}/ 3 + bx^{2}/ 2 + cx |
Quadratic inequalities.