Quadratic functions and equations.

Forms of equations of parabolas:
y=ax2 + bx + c
Ax2 + Ey + Dx + F = 0 is the standard form equation of a vertical parabola.
Completing-the-square form: a(x+b/2a)2+(c-b2/4a).
 or, a(x-h)2+k where (h,k) is the vertex, h being a horizontal shift, k a vertical shift.
x=ay2 + by + c is the equation for a horizontal parabola (which is not a function of x).
Ax2 + Bxy + Cy2 + Dx + Ey + F = 0, B2=4AC is the standard form equation of a general parabola, (one that can be oriented in any direction, which are not discussed here).
Only vertical parabolas represent functions.

ƒ(x) = ax2 + bx + c

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 symmetric 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 [even if only one or none].
The y-coordinate of the vertex is the function evaluated at this x:
    ƒ(-b/2a) = 4ac-b2/4a = c-b2/4a
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 ax2 + bx + c = 0. (i.e. y is 0)
There can be 0, 1 or 2 real number solutions:
b2-4ac is the discriminant which indicates how many (real number) zeroes/roots there are:
2 if the discriminant b2-4ac > 0. (b>√(ac)) x=(-b ± √(b2-4ac)) / 2a [quadratic formula]
1 if the discriminant b2-4ac = 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 b2-4ac < 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-b2+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.

Varying the coefficient a.
a>0: the function is opening upward. concave up
a<0: the function is opening downward. concave down.

The larger the |a|, the steeper the parabola, the faster it changes.
Varying the b coefficient.
Under construction...

Effect of varying the constant c:

roots Y intercept vertex area (to/from X axis) distance
mean value (to/from X axis) derivative integral ∫
0 0 (0,0) 0 0 0 2ax ax3/ 3
√(c/a) and -√(c/a) c (0,c) 4/3 c √(c/a) 2√(c/a) 2/3 c 2ax ax3/ 3 + cx
0 and -b/a 0 (-b/ 2a, -b2/ 4a) b3/ 6a2 b/a b2/ 6a 2ax+b ax3/ 3 + bx2/ 2
(-b ± √(b2-4ac)) / 2a c (-b/ 2a, c- b2/ 4a) √(b2-4ac)(4ac-b2) / 6a2 √(b2-4ac) / a 4ac-b2 / 6a 2ax+b ax3/ 3 + bx2/ 2 + cx

Area of the unique parabola that fits in a rectangle:

Quadratic inequalities.

Examples of quadratic functions