- Quadratic function
A quadratic function, in
mathematics, is a polynomial functionof the form , where . The graph of a quadratic function is a parabolawhose major axis is parallel to the "y"-axis.
The expression in the definition of a quadratic function is a polynomial of degree 2 or a 2nd degree polynomial, because the highest exponent of 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 or the zeros of the function.
Origin of word
In general, a prefix
quadr(i)-indicates the number 4. Examples are quadrilateraland quadrant. "Quadratum" is the Latin word for square because a square has four sides.
The two roots of the quadratic equation , where are
This formula is called the
* If , then there are two distinct roots since is a positive real number.
* If , then the two roots are equal, since is zero.
* If , then the two roots are
complex conjugates, since is imaginary.
By letting and or vice versa, one can factor as .
Forms of a quadratic function
A quadratic function can be expressed in three formats:
* is called the general form or polynomial form,
* is called the factored form, where and are the roots of the quadratic equation, it is used in
* is called the standard form or vertex form.
To convert the general form to factored form, one needs only the quadratic formula to determine the two roots and . To convert the general form to standard form, one needs a process called
completing the square. To convert the factored form (or standard form) to general form, one needs to multiply, expand and/or distribute the factors.
Regardless of the format, the graph of a quadratic function is a
parabola(as shown above).
* If , the parabola opens upward.
* If , 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).
The coefficient "b" alone is the declivity of the parabola as it crosses the y-axis.
The coefficient "c" controls the height of the parabola, more specifically, it is the point were the parabola crosses the "y"-axis.
The "x"-intercepts of the graph are the same as the roots of the quadratic function (see above).
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 standard form, the vertex is . By the method of completing the square, one can turn the general form: to
so the vertex of the parabola in the general form will be
If the quadratic function is in factored form
the average of the two roots, i.e.,
is the "x"-coordinate of the vertex, and hence the vertex is
The vertex is also the maximum point if or the minimum point if .
The vertical line
that passes through the vertex is also the axis of symmetry of the parabola.
*Maximum and minimum points
:The maximum or minimum of the function is always obtained at the vertex, the following method is another derivation of the same fact using
calculus, the advantage of this method is that it works for more general functions.
:Taking as sample quadratic equation, to find its maximum or minimum points (which depends on , if , it has a minimum point, if , it has a maximum point) we have to first, take its
:Then, we find the roots of :
:So, is the value of . Now, to find the value, we substitute on :
:Thus, the maximum or minimum point coordinates are:
The square root of a quadratic function
square rootof a quadratic function gives rise either to an ellipseor to a hyperbola.If then the equation describes a hyperbola. The axis of the hyperbola is determined by the ordinateof the minimumpoint of the corresponding parabola .
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 then the equation describes either an ellipse or nothing at all. If the ordinate of the
maximumpoint of the corresponding parabola is positive, then its square root describes an ellipse, but if the ordinate is negative then it describes an empty locus of points.
Bivariate quadratic function
A bivariate quadratic function is a second-degree polynomial of the form:Such a function describes a quadratic
surface. Setting equal to zero describes the intersection of the surface with the plane , which is a locus of points equivalent to a conic section.
If the function has no maximum or minimum, its graph forms an hyperbolic
If the function has a minimum if "A">0, and a maximum if "A"<0, its graph forms an elliptic
The minimum or maximum of a bivariate quadratic function is obtained at where:
If and the function has no maximum or minimum, its graph forms a parabolic cylinder.
If and 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.
Matrix representation of conic sections
Periodic points of complex quadratic mappings
List of mathematical functions
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