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Quadratic Functions

Definition

Definition Quadratic Function
A quadratic function is \(x \mapsto ax^2 + bx + c\) where \(a \neq 0\).
Example
For \(f(x) = x^2 - 3x + 1\), evaluate \(f(2)\).

\(\begin{aligned}[t]f(2) &= (2)^2 - 3(2) + 1 \\&= 4 - 6 + 1 \\&= -1\end{aligned}\)

Graph

The parabola is one of the conic sections, which are the group of curves obtained by intersecting a cone with a plane. A parabola is produced by intersecting the cone with a plane parallel to its generating line. By intersecting the cone at other angles, we can produce circles, hyperbolas, and ellipses.
Definition Parabola
Given a quadratic function \(x \mapsto ax^2 + bx + c\) where \(a \neq 0\), its graph is called a parabola.
Example
Sketch the graph of \(x \mapsto x^2 - x - 1\).

A table of values is
\(x\) \(-2\) \(-1\) 0 1 2
\(y\) 5 1 \(-1\) \(-1\) 1

To understand the concavity based on the sign of \(a\), use the GeoGebra animation at https://www.geogebra.org/m/gn3c2sqe.
Proposition Concavity
For any quadratic function \(x \mapsto ax^2 + bx + c\), \(a \neq 0\):
  • If \(a > 0\), the graph is concave up: .
  • If \(a < 0\), the graph is concave down: .

Solving \(f(x) \equal y\)

Method Solving \(f(x) \equal y\)
When solving for a value of \(f(x) = y\), we obtain a quadratic equation in \(x\). Since it is quadratic, there may be 0, 1, or 2 real solutions for \(x\).
Example
For \(f(x) = 2x^2 - 5x + 2\), find the \(x\)-intercepts of the function.

Set \(f(x) = 0\): \(2x^2 - 5x + 2 = 0\), with \(a=2\), \(b=-5\), \(c=2\).
  • \(\begin{aligned}[t]\Delta &= b^2 - 4ac \\&= (-5)^2 - 4(2)(2) \\&= 25 - 16 \\&= 9\end{aligned}\)
  • As \(\Delta > 0\), there are 2 distinct roots.
  • \(\begin{aligned}[t]x &= \frac{-b - \sqrt{\Delta}}{2a} &\text{ or } x &= \frac{-b + \sqrt{\Delta}}{2a} \\x &= \frac{-(-5) - \sqrt{9}}{2 \cdot 2} &\text{ or } x &= \frac{-(-5) + \sqrt{9}}{2 \cdot 2} \\x &= \frac{5 - 3}{4} &\text{ or } x &= \frac{5 + 3}{4} \\x &= \frac{2}{4} &\text{ or } x &= \frac{8}{4} \\x &= \frac{1}{2} &\text{ or } x &= 2\end{aligned}\)
The \(x\)-intercepts are at \(x = \frac{1}{2}\) and \(x = 2\).

Proposition Relative position between the graph and the \(x\)-axis
For any quadratic function \(x \mapsto ax^2 + bx + c\) and the discriminant \(\Delta = b^2 - 4ac\):
  • If \(\Delta > 0\), then the graph intersects the \(x\)-axis twice.
  • If \(\Delta = 0\), then the graph touches the \(x\)-axis at one point.
  • If \(\Delta < 0\), then the graph does not intersect the \(x\)-axis.