Quadratic function

A quadratic function ( quite rational function of second degree or second degree polynomial ) is a function that has as function term is a polynomial of degree 2, that of the form

Is. The graph is the parabola with the equation. For a linear function would.

  • 4.1 axis intersections
  • 4.2 vertex calculation using known zeros
  • 4.3 p- q- formula, discriminant and amount of solution
  • 4.4 The set of Vieta
  • 4.5 zeros and linear factors
  • 4.6 intersection of the parabola and a straight line
  • 4.7 Intersection of two parabolas

The general quadratic function

The correlation of the general quadratic function. Is and one obtains the square function. Your graph is the standard parabola. The coefficients, and determine the range of values ​​and the shape of the graph.

Parameters a

How is the value of the shape of the graph changes, can best be seen when and sets. One then obtains a stretched or compressed, and optionally reflected by the X- axis normal parabola.

For: the graph is simply mirrored on the x - axis compared to the normal parabola.

Yield at

Stretching at

Parameter c

A change of the parameter causes a shift in the y- direction. Is increased by one, then the graph is shifted by one unit to the above. Is decreased by one, the graph is, however, shifted one unit down.

Parameter b

The parameter specifies the slope of the parabola at the intersection with the y- axis. In particular, one can recognize the signs of whether the y- axis with the falling or the rising branch of the parabola is cut. From this it can in turn draw conclusions about the number and the possible location of zeros of pull.

A change of the parameter results in a shift in both the x - and y - direction. Is increased by one, then the graph is shifted to the left and down units. Is decremented by one, however, the graph is shifted to the right and above units.

Vertex determination

The vertex is decisive for the location of the parabola and represents either the absolute minimum (if positive ) or the absolute maximum (if negative). The coordinates of the vertex can be read directly if the function term is present in the vertex form:

The apex has the coordinates. The graph is axisymmetric with a line parallel to the y-axis. The vertex can be determined from the form shown by square addition.

Another way of calculating the vertex offers the calculus. Since the vertex always is a (local) point extreme ( maximum or minimum), the zero of the first derivative of the function returns the value of x of the vertex:

By insertion results in the y- value:

Determination of the vertex of the quadratic function.

  • Determination of the vertex on the vertex form of the function
  • Determination of the vertex with the help of differential calculus

Zeros of a quadratic function

The zeros of the quadratic function are obtained by solution of the equation, that is the quadratic equation

Illustrate the quadratic function by a conic

The graph of any quadratic function ( a parabola) is represented geometrically as the intersection of a plane with a cone. See advice at conic.

Focal point of the corresponding parabolic

The graph of a quadratic function is a parabola, and thus has a focal point. This is practically used in a parabolic mirror. With such a mirror can receive television programs or generate as high temperatures with solar energy. See also parable (mathematics).

The focus of the parabola with the equation.

Other properties of quadratic functions

Axis intersections

Vertex calculation using known zeros

, The zeros of the quadratic function is known, then the coordinates of the vertex can be calculated as follows:

P- q- formula, discriminant and amount of solution

The normal form of a quadratic equation is:

The set of Vieta

Are the solutions to the quadratic equation, then by the root set of Vieta and.

Zeros and linear factors

Sind and the zeros of the quadratic function, then we can write the functional equation as a product of its linear factors:

Intersection of the parabola and straight line

Is the functional equation of a parabola and the straight line a. Approach: equate the functional equations quadratic equation. Now, if:

Intersection of two parabolas

Are the functional equations of two parabolas. Approach: equate the functional equations quadratic equation. Now, if:

Reflection symmetry Conic section Focus (geometry)
355976
de