Properties of Quadratic Functions

Quadratic functions, characterized by a degree of 2, manifest as parabolic graphs without domain restrictions. Take, for instance, the quadratic function \(f(x)=2x^2-3x+1\). The graph of this function forms a parabola.

A critical feature of quadratic functions is the constant nonzero second differences when presented in a table of values. Let's illustrate this with the corrected table for the function \(f(x)=x^2+2x+3\):

Here, we observe that the second differences (i.e., the differences between consecutive function values) are consistent (1, 1, 1), affirming a quadratic relationship.

The parabolic nature of the graph is determined by the sign of the second differences. Consider the function \(g(x)=-x^2+4x-2\). As the second differences in the table are negative (-2, -2, -2), the parabola opens downward.

Quadratic functions can be expressed in various forms. The standard form is \(f(x)=ax^2+bx+c\), where \(a\), \(b\), and \(c\) are coefficients. For instance, if \(a\)=2, \(b\)=−3, and \(c\)=1, the standard form of \(f(x)\) is \(f(x)=2x^2−3x+1\).

Conversion between different forms involves algebraic techniques. To move from standard form to vertex form, completing the square can be employed. For factored form, the function can be expressed as a product of factors. For example, using \(f(x)=x^2+2x+3\), the factored form might be \(f(x)=(x+1)(x+3)\), where the roots of the quadratic equation provide the factors.

Vertex form is expressed as \(g(x)=a(x−h)^2+k\), with \((h,k)\) as the vertex of the parabola. If \(h=−1\) and \(k=4\), then the vertex form of \(g(x)\) is \(g(x)=(x+1)^2+4\).

Understanding these different forms equips us with a versatile toolkit for analyzing and solving problems involving quadratic functions, allowing us to represent them in ways most suitable for specific situations.