Quadratic function
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A quadratic function, in mathematics, is a polynomial function of the form [f(x)=ax^2+bx+c \,\!], where [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.
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 (which in turn is because a square has 4 sides,) and the area of a square is x^2, the prefix is also sometimes used in words involving the number 2, and this is an example.
Roots
The roots of the quadratic equation [f(x)=ax^2+bx+c \,\!], where [a \ne 0 \,\!] are
[ x = \frac} ]
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 [\Delta = b^2 - 4ac \,\!], 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 [\Delta > 0\,\!] and [\Delta] is a square number, then there are two distinct rational roots since [\sqrt] is rational.
- If [\Delta > 0\,\!] and [\Delta] is not a square number, then there are two distinct irrational roots since [\sqrt] is irrational.
- If [\Delta = 0\,\!], then there are two equal (a.k.a. double) roots since [\sqrt] is zero.
- If [\Delta < 0\,\!], then there are two distinct complex roots since [\sqrt] is imaginary.
By letting [ r_1 = \frac} ] and [ r_2 = \frac} ] or vice versa, one can factor [ a x^2 + b x + c \,\!] as [ a(x - r_1)(x - r_2)\,\!].
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,
- [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.
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.
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).
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\right)^2 - \frac ], so that the vertex of the parabola in the general form will be [ \left(-\frac, -\frac\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 \,\!], is the x-coordinate of the vertex, and hence the vertex is [ \left(\frac, f(\frac)\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]
- So, [-\frac ] is the [x\,\!] value of [f(x)\,\!]. Now, to find the [y\,\!] value, we substitute [x = -\frac ] on [f(x)\,\!]:
- [y=a \left (-\frac \right)^2+b \left (-\frac \right)+c\Rightarrow y= \frac - \frac + c \Rightarrow y= \frac - \frac + c \Rightarrow]
- [y= \frac \Rightarrow y= \frac \Rightarrow y= -\frac \Rightarrow y= -\frac ]
- Thus, the maximum or minimum point coordinates are:
- [ \left (-\frac , -\frac \right) ]
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 ]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 ] 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.
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 \,\!]
Minimum/Maximum
The minimum or maximum of a bivariate quadratic function is:
- [x_m = -\frac]
- [y_m = -\frac]
See also
External links
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