Quadratic function

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f(x) = x² - x - 2<center>

A quadratic function, in mathematics, is a polynomial function of the form <math>f(x)=ax^2+bx+c \,\!</math>, where <math>a, b, c \,\!</math> are real numbers and <math>a \ne 0 \,\!</math>. It takes its name from the Latin quadratus for square, because quadratic functions arise in the calculation of areas of squares. Because the (highest) exponent of x is 2, a quadratic function is sometimes referred as a degree 2 polynomial or a 2nd degree polynomial. 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 solutions to the equation are called the roots of the equation or the zeros of the function.

1 Origin of word
2 Roots
3 Forms of a quadratic function
4 Graph
- 4.1 Number of x-intercepts
- 4.2 Vertex
5 The square root of a quadratic function
6 Bivariate quadratic function
- 6.1 Minimum/Maximum
7 See also
8 External links

[edit] Origin of word

The prefix quadri- is used to indicate the number 4. Examples are quadrilateral and quadrant. However, because it is in the Latin word for square (since a square has 4 sides), and the area of a square with side length <math>x</math> is <math>x^2</math>, the prefix is also sometimes used in words involving the number 2.

[edit] Roots

The roots of the quadratic equation <math>0=ax^2+bx+c\,\!</math>, where <math>a \ne 0 \,\!</math> are

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

In the case where a, b and c are integers, the nature of the roots can be determined by the quantity <math>\Delta = b^2 - 4ac \,\!</math>, which is called the discriminant. In the case where a, b and c are rational, one can multiply a, b and c by their least common multiple to transform them to integers ( multiplying a nonzero constant to an equation will not change the roots nor their nature). In the case where a, b and c are real, the following does not always apply.

If <math>\Delta > 0\,\!</math> and <math>\Delta</math> is a square number, then there are two distinct rational roots since <math>\sqrt{\Delta}</math> is rational.
If <math>\Delta > 0\,\!</math> and <math>\Delta</math> is not a square number, then there are two distinct irrational roots since <math>\sqrt{\Delta}</math> is irrational.
If <math>\Delta = 0\,\!</math>, then there are two equal (a.k.a. double) roots since <math>\sqrt{\Delta}</math> is zero.
If <math>\Delta < 0\,\!</math>, then there are two distinct complex roots since <math>\sqrt{\Delta}</math> is imaginary.

By letting <math> r_1 = \frac{-b + \sqrt{b^2 - 4 a c}}{2 a} </math> and <math> r_2 = \frac{-b - \sqrt{b^2 - 4 a c}}{2 a} </math> or vice versa, one can factor <math> a x^2 + b x + c \,\!</math> as <math> a(x - r_1)(x - r_2)\,\!</math>.

[edit] Forms of a quadratic function

A quadratic function can be expressed in three formats:

<math>f(x) = a x^2 + b x + c \,\!</math> is called the general form or polynomial form,
<math>f(x) = a(x - r_1)(x - r_2) \,\!</math> is called the factored form, where <math> r_1 </math> and <math> r_2 </math> are the roots of the quadratic equation, and
<math>f(x) = a(x - h)^2 + k \,\!</math> 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 <math> r_1 </math> and <math> r_2 </math>. 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.

[edit] Graph

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

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

[edit] Number of x-intercepts

The number of x-intercepts can be determined by the discriminant too.

If <math>\Delta > 0\,\!</math>, then there are two x-intercepts because the two real roots are distinct.
If <math>\Delta = 0\,\!</math>, then there is exactly one x-intercept because of the two real roots are equal. In this case, the parabola is tangent to the x-axis.
If <math>\Delta < 0\,\!</math>, the graph has no x-intercepts because the two roots are imaginary. In this case, the parabola is either completely above the x-axis (if a > 0) or completely below the x-axis (if a < 0).

[edit] 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 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> If the quadratic function is in factored form <math>f(x) = a(x - r_1)(x - r_2) \,\!</math>, the average of the two roots, i.e., <math>\frac{r_1 + r_2}{2} \,\!</math>, is the x-coordinate of the vertex, and hence the vertex is <math> \left(\frac{r_1 + r_2}{2}, f(\frac{r_1 + r_2}{2})\right)\!</math>. The vertex is also the maximum point if <math>a < 0 \,\!</math> or the minimum point if <math>a > 0 \,\!</math>.

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>

[edit] 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 <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.

[edit] Bivariate quadratic function

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

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.