# quadratic function

A quadratic function is a function expressed by a polynomial of degree two.

## Overview

What is a quadratic function
f
(
x
)
a
x
2
+
b
x
+
c
(
a
≠
0
)
{\displaystyle f(x)ax^{2}+bx+c\quad (a\neq 0)}
It is a function expressed in the form of If the coefficients a, b, and c are real-valued constants and x is a real-valued variable, the graph is a parabola in the xy-coordinate system.
In this section, we focus on quadratic functions as real-valued functions, and describe well-known matters in analytic geometry.

## definition

function defined by a polynomial of degree 2
f
(
x
)
a
x
2
+
b
x
+
c
(
a
≠
0
)
{\displaystyle f(x)ax^{2}+bx+c\quad (a\neq 0)}
is called a quadratic function with x as the independent variable. In particular, when b c 0, it is also called "function proportional to the square".
f
(
x
)
a
(
x
+
b
2
a
)
2
−
b
2
−
Four
a
c
Four
a
{\displaystyle f(x)a\left(x+{\frac {b}{2a}}\right)^{2}-{\frac {b^{2}-4ac}{4a}}}
In the normal form above, the coordinates of the vertices of the quadratic function are generally
(
x
,
y
)
(
−
b
2
a
,
−
b
2
−
Four
a
c
Four
a
)
{\displaystyle (x,\,y)\left(-{\frac {b}{2a}},\,-{\frac {b^{2}-4ac}{4a}}\right)}
becomes.
A quadratic function expressed in the form f(x) ax2 + bx + c is called a general form (standard form). A function that can be transformed into a general form by transformation is also called a quadratic function, especially
A quadratic function of the form f(x) a(x - p)2 + q is called the standard form (vertex form).
A quadratic function of the form f(x) a(x - s)(x - t) is called a factored form, or simply a factored form.
It is also the normal form when b 0 in the general form, and the factored form when q 0 in the normal form. It is also the normal form when s t in the factored form, and the general form when s t 0.
Expanding the normal or factored form gives the general form, and factoring the general form gives the factored form. Also, if you complete the square of the general form, you will get the standard form.
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