Quadratic function

From Wikipedia, the free encyclopedia

f(x) = x² - x - 2

A quadratic function, in mathematics, is a polynomial function of the form $f(x)=ax^2+bx+c \,\!$ , where $a, b, c \,\!$ are real numbers and $a \ne 0 \,\!$ . 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 $x$ is $x 2$ , the prefix is also sometimes used in words involving the number 2.

[edit] Roots

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

$x = \frac{-b \pm \sqrt{b^2 - 4 a c}}{2 a}$

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

say $Δ = b 2 - 4 a c$
If $\Delta > 0\,\!$ and $Δ$ is a square number, then there are two distinct rational roots since $\sqrt{\Delta}$ is rational.
If $\Delta > 0\,\!$ and $Δ$ is not a square number, then there are two distinct irrational roots since $\sqrt{\Delta}$ is irrational.
If $\Delta = 0\,\!$ , then there are two equal (a.k.a. double) roots since $\sqrt{\Delta}$ is zero.
If $\Delta < 0\,\!$ , then there are two distinct complex roots since $\sqrt{\Delta}$ is imaginary.

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

[edit] 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 general form or polynomial form,
$f(x) = a(x - r_1)(x - r_2) \,\!$ is called the factored form, where $r 1$ and $r 2$ are the roots of the quadratic equation, and
$f(x) = a(x - h)^2 + k \,\!$ 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 $r 1$ and $r 2$ . 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 $a > 0 \,\!$ , the parabola opens upward.
If $a < 0 \,\!$ , the parabola opens downward.

[edit] Number of x-intercepts

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

If $\Delta > 0\,\!$ , then there are two x-intercepts because the two real roots are distinct.
If $\Delta = 0\,\!$ , 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 $\Delta < 0\,\!$ , 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 $(h, k)\,\!$ . By the method of completing the square, one can turn the general form $f(x) = a x^2 + b x + c \,\!$ to $f(x) = a\left(x + \frac{b}{2a}\right)^2 - \frac{b^2-4ac}{4 a}$ , so that the vertex of the parabola in the general form will be $\left(-\frac{b}{2a}, -\frac{\Delta}{4 a}\right).$ 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 $\left(\frac{r_1 + r_2}{2}, f(\frac{r_1 + r_2}{2})\right)\!$ . The vertex is also the maximum point if $a < 0 \,\!$ or the minimum point if $a > 0 \,\!$ .

Maximum and minimum points

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

$f(x)=ax^2+bx+c \Leftrightarrow \,\!$ $f'(x)=2ax+b \,\!$

Then, we find the root of $f'(x)\,\!$ :

$2ax+b=0 \Rightarrow \,\!$ $2ax=-b \Rightarrow\,\!$ $x=-\frac{b}{2a}$

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

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

$y= \frac{b^2 - 2b^2 + 4ac}{4a} \Rightarrow y= \frac{-b^2+4ac}{4a} \Rightarrow y= -\frac{(b^2-4ac)}{4a} \Rightarrow y= -\frac{\Delta}{4a}$

Thus, the maximum or minimum point coordinates are:

$\left (-\frac {b}{2a}, -\frac {\Delta}{4a} \right)$

[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 $a>0\,\!$ 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.

[edit] Bivariate 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.