Quadratic function

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


 * $$f(x)=ax^2+bx+c,\quad a \ne 0.$$

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

The expression $$ax^2+bx+c$$ 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
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
The roots (zeros) of the quadratic function
 * $$f(x) = ax^2+bx+c\,$$

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

When the coefficients a, b, and c, are real or complex, the roots are
 * $$x=\frac{-b \pm \sqrt{\Delta}}{2 a}, $$

where the discriminant is defined as
 * $$\Delta = b^2 - 4 a c \, . $$

Forms of a quadratic function
A quadratic function can be expressed in three formats:


 * $$f(x) = a x^2 + b x + c \,\!$$ is called the standard form,
 * $$f(x) = a(x - x_1)(x - x_2)\,\!$$ is called the factored form, where $$ x_1 $$ and $$ x_2 $$ are the roots of the quadratic equation, it is used in logistic map
 * $$f(x) = a(x - h)^2 + k \,\!$$ 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 $$ x_1 $$ and $$ x_2 $$. 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
Regardless of the format, the graph of a quadratic function is a parabola (as shown above).
 * If $$a > 0 \,\!$$ (or is a positive number), the parabola opens upward.
 * If $$a < 0 \,\!$$ (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 $$x = -\frac{b}{2a}$$.

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
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 $$(h, k)\,\!$$. By the method of completing the square, one can turn the general form
 * $$f(x) = a x^2 + b x + c \,\!$$

into
 * $$ f(x) = a\left(x + \frac{b}{2a}\right)^2 - \frac{b^2-4ac}{4 a} ,$$

so the vertex of the parabola in the vertex form is
 * $$ \left(-\frac{b}{2a}, -\frac{\Delta}{4 a}\right). $$

If the quadratic function is in factored form
 * $$f(x) = a(x - r_1)(x - r_2) \,\!$$

the average of the two roots, i.e.,
 * $$\frac{r_1 + r_2}{2} \,\!$$

is the x-coordinate of the vertex, and hence the vertex is
 * $$ \left(\frac{r_1 + r_2}{2}, f\left(\frac{r_1 + r_2}{2}\right)\right).\!$$

The vertex is also the maximum point if $$a < 0 \,\!$$ or the minimum point if $$a > 0 \,\!$$.

The vertical line


 * $$ x=h=-\frac{b}{2a} $$

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

Maximum and minimum points
Using calculus, the vertex point, being a maximum or minimum of the function, can be obtained by finding the roots of the derivative:
 * $$f(x)=ax^2+bx+c \quad \Rightarrow \quad f'(x)=2ax+b \,\!,$$

giving
 * $$x=-\frac{b}{2a}$$

with the corresponding function value
 * $$f(x) = a \left (-\frac{b}{2a} \right)^2+b \left (-\frac{b}{2a} \right)+c = -\frac{(b^2-4ac)}{4a} = -\frac{\Delta}{4a} \,\!,$$

so again the vertex point coordinates can be expressed as
 * $$ \left (-\frac {b}{2a}, -\frac {\Delta}{4a} \right). $$

The square root of a quadratic function
The square root of a quadratic function gives rise either to an ellipse or to a hyperbola.If $$a>0\,\!$$ then the equation $$ y = \pm \sqrt{a x^2 + b x + c} $$ describes a hyperbola. The axis of the hyperbola is determined by the ordinate of the minimum point of the corresponding parabola $$ y_p = a x^2 + b x + c \,\!$$. 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 $$a<0\,\!$$ then the equation $$ y = \pm \sqrt{a x^2 + b x + c} $$ describes either an ellipse or nothing at all. If the ordinate of the maximum point of the corresponding parabola $$ y_p = a x^2 + b x + c \,\!$$ is positive, then its square root describes an ellipse, but if the ordinate is negative then it describes an empty locus of points.

Iteration
Given an $$f(x)=ax^2+bx+c$$, one cannot always deduce the analytic form of $$f^{(n)}(x)$$, which means the nth iteration of $$f(x)$$. (The superscript can be extended to negative number referring to the iteration of the inverse of $$f(x)$$ if the inverse exists.) But there is one easier case, in which $$f(x)=a(x-x_0)^2+x_0$$.

In such case, one has
 * $$f(x)=a(x-x_0)^2+x_0=h^{(-1)}(g(h(x)))\,\!$$,

where
 * $$g(x)=ax^2\,\!$$ and $$h(x)=x-x_0\,\!$$.

So by induction,
 * $$f^{(n)}(x)=h^{(-1)}(g^{(n)}(h(x)))\,\!$$

can be obtained, where $$g^{(n)}(x)$$ can be easily computed as
 * $$g^{(n)}(x)=a^{2^{n}-1}x^{2^{n}}\,\!$$.

Finally, we have
 * $$f^{(n)}(x)=a^{2^n-1}(x-x_0)^{2^n}+x_0\,\!$$,

in the case of $$f(x)=a(x-x_0)^2+x_0$$.

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
A bivariate quadratic function is a second-degree polynomial of the form
 * $$ f(x,y) = A x^2 + B y^2 + C x + D y + E x y + F \,\!$$

Such a function describes a quadratic surface. Setting $$f(x,y)\,\!$$ equal to zero describes the intersection of the surface with the plane $$z=0\,\!$$, which is a locus of points equivalent to a conic section.

Minimum/maximum
If $$ 4AB-E^2 <0 \,$$ the function has no maximum or minimum, its graph forms an hyperbolic paraboloid.

If $$ 4AB-E^2 >0 \,$$ 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 $$ (x_m, y_m) \,$$ where:


 * $$x_m = -\frac{2BC-DE}{4AB-E^2}$$


 * $$y_m = -\frac{2AD-CE}{4AB-E^2}$$

If $$ 4AB- E^2 =0 \,$$ and $$ DE-2CB=2AD-CE \ne 0 \,$$ the function has no maximum or minimum, its graph forms a parabolic cylinder.

If $$ 4AB- E^2 =0 \,$$ and $$ DE-2CB=2AD-CE =0 \,$$ 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.