A quadratic function is given by

\[f(x) = ax^2 + bx + c\]

where \(a \neq 0\). The graph of a quadratic function is called a parabola such that:

  • it always has a cup-shaped curve
  • it opens upward if \(a > 0\) or opens downwards if \(a < 0\)
  • it has a turning point, or vertex, whose coordinate is \(x = - \frac{b}{2a}\)
  • the vertical line \(x = \frac{-b}{2a}\) (which is not part of the graph) is the line of symmetry

You could think of the vertex as the first \(y\)-value. The \(x\)-intercept(s) are the \(y\)-value(s) where \(x = 0\).

The solutions of any quadratic equation \(ax^2 + bx + c = 0\), with \(a \neq 0\), are given by the quadratic formula

\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]

When \(\sqrt{b^2 - 4ac} < 0\) there are no \(x\)-intercepts (not real-valued solutions). There are solutions in the complex plane. The standard form of quadratic equations hints at the fact that solving them means finding what are called their “roots”, that’s where their value is equal to zero. Being the graph of a quadratic equation a parabola, it’s then clear why we might have one, two, or no solutions (which would correspond to the \(x\)-intercepts, depending on the position of the parabola).

Let’s explore the function \(f(x) = x^2 - 2x - 3\).

import matplotlib.pyplot as plt
import numpy as np

def quadratic(x):
  return x ** 2 - 2 * x - 3

yvalues = [quadratic(x) for x in np.linspace(-8, 10)]

xvertex = - (-2 / 2 * 1)
xvertex # which is also the vertical line of symmetry
## 1.0
yvertex = quadratic(xvertex)
## -4.0
plt.plot(range(-25, 25), yvalues)