Quadratic function

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Image:Polynomialdeg2.png A quadratic function, in mathematics, is a polynomial function of the form <math>f(x)=ax^2+bx+c \,\!</math>, where <math>a\,\!</math> is nonzero. It takes its name from the Latin quadratus for square, because quadratic functions arise in the calculation of areas of squares. In the case where the domain and codomain are <math>\mathbb{R}</math> (all real numbers), the graph of such a function is a parabola.
If the quadratic function is set to be equal to zero, then the result is a quadratic equation.
The square root of a quadratic function gives rise either to an ellipse or to a hyperbola.
If <math>a>0\,\!</math> then the equation<math> y = \pm \sqrt{a x^2 + b x + c} </math>describes a hyperbola. The axis of the hyperbola is determined by the ordinate of the minimum point of the corresponding parabola<math> y_p = a x^2 + b x + c \,\!</math>
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 <math>a<0\,\!</math> then the equation <math> y = \pm \sqrt{a x^2 + b x + c} </math> describes either an ellipse or nothing at all. If the ordinate of the maximum point of the corresponding parabola <math> y_p = a x^2 + b x + c \,\!</math> is positive, then its square root describes an ellipse, but if the ordinate is negative then it describes an empty locus of points.

Contents 1 Roots 2 Graph 2.1 Vertex 2.2 Number of x-intercepts 3 Bivariate quadratic function 3.1 Minimum/Maximum 4 See also

Roots

The roots, or solutions to the quadratic function, for variable <math>x\,\!</math>, are

<math> x = \frac{-b \pm \sqrt{b^2 - 4 a c}}{2 a} </math>

This formula is called the quadratic formula. To see how the formula is derived, see quadratic equation.

One can also factor quadratics to derive the two base polynomial functions that make up each quadratic equation.

Graph

The graph of a quadratic function <math>f(x) = a x^2 + b x + c \,\!</math> or <math>f(x) = a(x - h)^2 + k \,\!</math> is called a parabola.

The former is called the general form while the latter is the standard form. In either form, <math>a</math> is non-zero, and

• If <math>a > 0 \,\!</math>, the parabola opens upward.
• If <math>a < 0 \,\!</math>, the parabola opens downward.

Vertex

The place where the parabola turns is called the turning point or the vertex of the parabola. If the quadratic function is in standard form, the vertex is <math>(h, k)\,\!</math>. By the method of completing the square, one can turn the general form <math>f(x) = a x^2 + b x + c \,\!</math> to <math> f(x) = a\left(x + \frac{b}{2a}\right)^2 - \frac{b^2-4ac}{4 a} </math>, so that the vertex of the parabola in the general form will be <math> \left(-\frac{b}{2a}, -\frac{\Delta}{4 a}\right). </math> The vertex is also the maximum (if <math>a < 0 \,\!</math>) or the minimum (if <math>a > 0 \,\!</math>) point.

• Maximum and minimum points

Taking <math>f(x) = ax^2 + bx + c \,\!</math> as sample quadratic equation, to find its maximum or minimum points (which depends on <math>a \,\!</math>, if <math>a > 0 \,\!</math>, it has a minimum point, if <math>a < 0\,\!</math>, it has a maximum point) we have to, first, take its derivative:

<math>f(x)=ax^2+bx+c \Leftrightarrow \,\!</math><math>f'(x)=2ax+b \,\!</math>

Then, we find the root of <math>f'(x)\,\!</math>:

<math>2ax+b=0 \Rightarrow \,\!</math> <math>2ax=-b \Rightarrow\,\!</math> <math>x=-\frac{b}{2a}</math>

So, <math>-\frac{b} {2a}</math> is the <math>x\,\!</math> value of <math>f(x)\,\!</math>. Now, to find the <math>y\,\!</math> value, we substitute <math>x = -\frac{b} {2a}</math> on <math>f(x)\,\!</math>:

<math>y=a \left (-\frac{b}{2a} \right)^2+b \left (-\frac{b}{2a} \right)+c\Rightarrow y= \frac{ab^2}{4a^2} - \frac{b^2}{2a} + c \Rightarrow y= \frac{b^2}{4a} - \frac{b^2}{2a} + c \Rightarrow</math>

<math>y= \frac{b^2 - 2b^2 + 4ac}{4a} \Rightarrow y= \frac{-b^2+4ac}{4a} \Rightarrow y= -\frac{(b^2-4ac)}{4a} \Rightarrow y= -\frac{\Delta}{4a} </math>

Thus, the maximum or minimum point coordinates are:

<math> \left (-\frac {b}{2a}, -\frac {\Delta}{4a} \right) </math>

Number of x-intercepts

The number of x-intercepts is determined by the quantity <math>\Delta = b^2 - 4ac \,\!</math>, which is called the discriminant.

• If <math>\Delta > 0\,\!</math> and <math>\Delta</math> is a square number, then the graph has two rational x-intercepts since the quadratic formula yields two distinct real roots
• If <math>\Delta > 0\,\!</math>, and <math>\Delta</math> is not a square number, then the graph has two irrational x-intercepts since the quadratic formula yields two distinct real roots.
• If <math>\Delta = 0\,\!</math>, the graph has one x-intercept since the quadratic formula yields one real root (or two equal real roots).
• If <math>\Delta < 0\,\!</math>, the graph has no x-intercepts since the quadratic formula yields two imaginary roots.

Bivariate quadratic function

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

<math> f(x,y) = A x^2 + B y^2 + C x + D y + E x y + F \,\!</math>

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

Minimum/Maximum

The minimum or maximum of a bivariate quadratic function is:

<math>x_m = -\frac{2BC-DE}{4AB-E^2}</math>

<math>y_m = -\frac{2AD-CE}{4AB-E^2}</math>

See also

• quadratic form
• Matrix representation of conic sections
• quadric
• Quadratic Equation Solver
• A free program to solve quadratic equations. (v2.0)

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