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.
The quadratic function given by
$$
f(x) = a(x-h)^2 + k, \quad a \not = 0
$$
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 \(\displaystyle f \left( -\frac{b}{2a} \right). \)

E 2.1 Exercises