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Failed to parse (Missing texvc executable; please see math/README to configure.): x^2 - x - 2\!

A quadratic function, in mathematics, is a polynomial function of the form

Failed to parse (Missing texvc executable; please see math/README to configure.): f(x)=ax^2+bx+c,\quad a \ne 0.

The graph of a quadratic function is a parabola whose axis of symmetry is parallel to the y-axis.

The expression Failed to parse (Missing texvc executable; please see math/README to configure.): ax^2+bx+c

in the definition of a quadratic function is a polynomial of degree 2 or second order, or a 2nd degree polynomial, because the highest exponent of x is 2.

If the quadratic function is set equal to zero, then the result is a quadratic equation. The solutions to the equation are called the roots of the equation.

Origin of word[link]

The adjective quadratic comes from the Latin word quadrātum (“square”). A term like x² is called a square in algebra because it is the area of a square with side x.

In general, a prefix quadr(i)- indicates the number 4. Examples are quadrilateral and quadrant. Quadratum is the Latin word for square because a square has four sides.

Roots[link]

The roots (zeros) of the quadratic function

Failed to parse (Missing texvc executable; please see math/README to configure.): f(x) = ax^2+bx+c\,

are the values of x for which f(x) = 0.

When the coefficients a, b, and c, are real or complex, the roots are

Failed to parse (Missing texvc executable; please see math/README to configure.): x=\frac{-b \pm \sqrt{\Delta}}{2 a},

where the discriminant is defined as

Failed to parse (Missing texvc executable; please see math/README to configure.): \Delta = b^2 - 4 a c \, .

Forms of a quadratic function[link]

A quadratic function can be expressed in three formats:^[1]

• Failed to parse (Missing texvc executable; please see math/README to configure.): f(x) = a x^2 + b x + c \,\!

is called the standard form,

• Failed to parse (Missing texvc executable; please see math/README to configure.): f(x) = a(x - x_1)(x - x_2)\,\!

is called the factored form, where Failed to parse (Missing texvc executable; please see math/README to configure.):  x_1 
and Failed to parse (Missing texvc executable; please see math/README to configure.):  x_2 
are the roots of the quadratic equation, it is used in logistic map

• Failed to parse (Missing texvc executable; please see math/README to configure.): f(x) = a(x - h)^2 + k \,\!

is called the vertex form, where h and k are the x and y coordinates of the vertex, respectively.

To convert the standard form to factored form, one needs only the quadratic formula to determine the two roots Failed to parse (Missing texvc executable; please see math/README to configure.): x_1

and Failed to parse (Missing texvc executable; please see math/README to configure.):  x_2 

. To convert the standard form to vertex form, one needs a process called completing the square. To convert the factored form (or vertex form) to standard form, one needs to multiply, expand and/or distribute the factors.

Graph[link]

Failed to parse (Missing texvc executable; please see math/README to configure.): f(x) = ax^2 |_{a=\{0.1,0.3,1,3\}} \!

Failed to parse (Missing texvc executable; please see math/README to configure.): f(x) = x^2 + bx |_{b=\{1,2,3,4\}} \!

Failed to parse (Missing texvc executable; please see math/README to configure.): f(x) = x^2 + bx |_{b=\{-1,-2,-3,-4\}} \!

Regardless of the format, the graph of a quadratic function is a parabola (as shown above).

• If Failed to parse (Missing texvc executable; please see math/README to configure.): a > 0 \,\!

(or is a positive number), the parabola opens upward.

• If Failed to parse (Missing texvc executable; please see math/README to configure.): a < 0 \,\!

(or is a negative number), the parabola opens downward.

The coefficient a controls the speed of increase (or decrease) of the quadratic function from the vertex, bigger positive a makes the function increase faster and the graph appear more closed.

The coefficients b and a together control the axis of symmetry of the parabola (also the x-coordinate of the vertex) which is at Failed to parse (Missing texvc executable; please see math/README to configure.): x = -\frac{b}{2a} .

The coefficient b alone is the declivity of the parabola as y-axis intercepts.

The coefficient c controls the height of the parabola, more specifically, it is the point where the parabola intercept the y-axis.

Vertex[link]

The vertex of a parabola is the place where it turns, hence, it's also called the turning point. If the quadratic function is in vertex form, the vertex is Failed to parse (Missing texvc executable; please see math/README to configure.): (h, k)\,\! . By the method of completing the square, one can turn the general form

Failed to parse (Missing texvc executable; please see math/README to configure.): f(x) = a x^2 + b x + c \,\!

into

Failed to parse (Missing texvc executable; please see math/README to configure.): f(x) = a\left(x + \frac{b}{2a}\right)^2 - \frac{b^2-4ac}{4 a} ,

so the vertex of the parabola in the vertex form is

Failed to parse (Missing texvc executable; please see math/README to configure.): \left(-\frac{b}{2a}, -\frac{\Delta}{4 a}\right).

If the quadratic function is in factored form

Failed to parse (Missing texvc executable; please see math/README to configure.): f(x) = a(x - r_1)(x - r_2) \,\!

the average of the two roots, i.e.,

Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{r_1 + r_2}{2} \,\!

is the x-coordinate of the vertex, and hence the vertex is

Failed to parse (Missing texvc executable; please see math/README to configure.): \left(\frac{r_1 + r_2}{2}, f\left(\frac{r_1 + r_2}{2}\right)\right).\!

The vertex is also the maximum point if Failed to parse (Missing texvc executable; please see math/README to configure.): a < 0 \,\!

or the minimum point if Failed to parse (Missing texvc executable; please see math/README to configure.): a > 0 \,\!

.

The vertical line

Failed to parse (Missing texvc executable; please see math/README to configure.): x=h=-\frac{b}{2a}

that passes through the vertex is also the axis of symmetry of the parabola.

Maximum and minimum points[link]

Using calculus, the vertex point, being a maximum or minimum of the function, can be obtained by finding the roots of the derivative:

Failed to parse (Missing texvc executable; please see math/README to configure.): f(x)=ax^2+bx+c \quad \Rightarrow \quad f'(x)=2ax+b \,\!,

giving

Failed to parse (Missing texvc executable; please see math/README to configure.): x=-\frac{b}{2a}

with the corresponding function value

Failed to parse (Missing texvc executable; please see math/README to configure.): f(x) = a \left (-\frac{b}{2a} \right)^2+b \left (-\frac{b}{2a} \right)+c = -\frac{(b^2-4ac)}{4a} = -\frac{\Delta}{4a} \,\!,

so again the vertex point coordinates can be expressed as

Failed to parse (Missing texvc executable; please see math/README to configure.): \left (-\frac {b}{2a}, -\frac {\Delta}{4a} \right).

The square root of a quadratic function[link]

The square root of a quadratic function gives rise either to an ellipse or to a hyperbola.If Failed to parse (Missing texvc executable; please see math/README to configure.): a>0\,\!

then the equation Failed to parse (Missing texvc executable; please see math/README to configure.):  y = \pm \sqrt{a x^2 + b x + c} 
describes a hyperbola. The axis of the hyperbola is determined by the ordinate of the minimum point of the corresponding parabola Failed to parse (Missing texvc executable; please see math/README to configure.):  y_p = a x^2 + b x + c \,\!

.
If the ordinate is negative, then the hyperbola's axis is horizontal. If the ordinate is positive, then the hyperbola's axis is vertical.
If Failed to parse (Missing texvc executable; please see math/README to configure.): a<0\,\!

then the equation Failed to parse (Missing texvc executable; please see math/README to configure.):  y = \pm \sqrt{a x^2 + b x + c} 
describes either an ellipse or nothing at all.  If the ordinate of the maximum point of the corresponding parabola

Failed to parse (Missing texvc executable; please see math/README to configure.): y_p = a x^2 + b x + c \,\!

is positive, then its square root describes an ellipse, but if the ordinate is negative then it describes an empty locus of points.

Iteration[link]

Given an Failed to parse (Missing texvc executable; please see math/README to configure.): f(x)=ax^2+bx+c , one cannot always deduce the analytic form of Failed to parse (Missing texvc executable; please see math/README to configure.): f^{(n)}(x) , which means the nth iteration of Failed to parse (Missing texvc executable; please see math/README to configure.): f(x) . (The superscript can be extended to negative number referring to the iteration of the inverse of Failed to parse (Missing texvc executable; please see math/README to configure.): f(x)

if the inverse exists.) But there is one easier case, in which Failed to parse (Missing texvc executable; please see math/README to configure.): f(x)=a(x-x_0)^2+x_0

.

In such case, one has

Failed to parse (Missing texvc executable; please see math/README to configure.): f(x)=a(x-x_0)^2+x_0=h^{(-1)}(g(h(x)))\,\!

, where

Failed to parse (Missing texvc executable; please see math/README to configure.): g(x)=ax^2\,\!

and Failed to parse (Missing texvc executable; please see math/README to configure.): h(x)=x-x_0\,\!

. So by induction,

Failed to parse (Missing texvc executable; please see math/README to configure.): f^{(n)}(x)=h^{(-1)}(g^{(n)}(h(x)))\,\!

can be obtained, where Failed to parse (Missing texvc executable; please see math/README to configure.): g^{(n)}(x)

can be easily computed as

Failed to parse (Missing texvc executable; please see math/README to configure.): g^{(n)}(x)=a^{2^{n}-1}x^{2^{n}}\,\!

. Finally, we have

Failed to parse (Missing texvc executable; please see math/README to configure.): f^{(n)}(x)=a^{2^n-1}(x-x_0)^{2^n}+x_0\,\!

, in the case of Failed to parse (Missing texvc executable; please see math/README to configure.): f(x)=a(x-x_0)^2+x_0 .

See Topological conjugacy for more detail about such relationship between f and g. And see Complex quadratic polynomial for the chaotic behavior in the general iteration.

Bivariate (two variable) quadratic function[link]

A bivariate quadratic function is a second-degree polynomial of the form

Failed to parse (Missing texvc executable; please see math/README to configure.): f(x,y) = A x^2 + B y^2 + C x + D y + E x y + F \,\!

Such a function describes a quadratic surface. Setting Failed to parse (Missing texvc executable; please see math/README to configure.): f(x,y)\,\!

equal to zero describes the intersection of the surface with the plane Failed to parse (Missing texvc executable; please see math/README to configure.): z=0\,\!

, which is a locus of points equivalent to a conic section.

Minimum/maximum[link]

If Failed to parse (Missing texvc executable; please see math/README to configure.): 4AB-E^2 <0 \,

the function has no maximum or minimum, its graph forms an hyperbolic paraboloid.

If Failed to parse (Missing texvc executable; please see math/README to configure.): 4AB-E^2 >0 \,

the function has a minimum if A>0, and a maximum if A<0, its graph forms an elliptic paraboloid.

The minimum or maximum of a bivariate quadratic function is obtained at Failed to parse (Missing texvc executable; please see math/README to configure.): (x_m, y_m) \,

where:

Failed to parse (Missing texvc executable; please see math/README to configure.): x_m = -\frac{2BC-DE}{4AB-E^2}

Failed to parse (Missing texvc executable; please see math/README to configure.): y_m = -\frac{2AD-CE}{4AB-E^2}

If Failed to parse (Missing texvc executable; please see math/README to configure.): 4AB- E^2 =0 \,

and Failed to parse (Missing texvc executable; please see math/README to configure.):  DE-2CB=2AD-CE \ne 0 \,
the function has no maximum or minimum, its graph forms a parabolic cylinder.

If Failed to parse (Missing texvc executable; please see math/README to configure.): 4AB- E^2 =0 \,

and Failed to parse (Missing texvc executable; please see math/README to configure.):  DE-2CB=2AD-CE =0 \,
the function achieves the maximum/minimum at a line. Similarly, a minimum if A>0 and a maximum if A<0, its graph forms a parabolic cylinder.

See also[link]

• Quadratic form
• Quadratic equation
• Quadratic polynomial
• Matrix representation of conic sections
• Quadric
• Periodic points of complex quadratic mappings
• List of mathematical functions

References[link]

External links[link]

• Weisstein, Eric W., "Quadratic" from MathWorld.