# 2.1 | Quadratic Functions and Models

Let $$n$$ be a nonnegative integer and let $$a_n, a_{n-1}, \ldots, a_2, a_1, a_0$$ be real numbers with $$a_n \not = 0.$$ The function given by
$$f(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_2 x^2 + a_1 x + a_0$$
is called a polynomial function of x with degree $$n$$. When $$n = 0,1,2,3,4,$$ or $$5$$, we call $$f$$ a constant, linear, quadratic, cubic, quartic, or quintic function, respectively.
$$f(x) = a(x-h)^2 + k, \quad a \not = 0$$
is in standard form. The graph of $$f$$ is a parabola whose axis is the vertical line $$x = h$$ and whose vertex is the point $$(h,k).$$ If $$a >0,$$ the parabola opens upward, and if $$a<0,$$ the parabola opens downward.
Consider the function $$f(x) = ax^2 + bx + c$$ with vertex
$$\left(-\frac{b}{2a}, f \left(- \frac{b}{2a} \right) \right).$$

1. If $$a>0$$, $$f$$ has a minimum at $$\displaystyle x = – \frac{b}{2a}.$$ The minimum value is $$\displaystyle f \left( -\frac{b}{2a} \right).$$
2. If $$a<0$$, $$f$$ has a maximum at $$\displaystyle x = - \frac{b}{2a}.$$ The maximum value is $$\displaystyle f \left( -\frac{b}{2a} \right).$$