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Quadratic functions

A quadratic function takes the form $$$ y = \Tred{a}x^2 + \Tgreen{b}x + \Tblue{c}$$$ for three numbers $$\Tred{a}\ne0$$, $$\Tgreen{b}$$ and $$\Tblue{c}$$.

$$y = \Tred{-2} x^2 + \Tblue{1}$$ and $$y = \Tred{2}x^2 \Tgreen{-3} x - \Tblue{6}$$ are quadratic.

Constant functions and linear functions are NOT quadratic because they correspond to $$\Tred{a}=0$$.

The functions $$x^3$$ and $$1/x$$ are not quadratic.

  • If $$\Tred{a}\gt 0$$, the graph has a U shape. It is symmetrical across the vertical line through its minimum point.
  • If $$\Tred{a}\lt 0$$, the graph has an inverted U shape. It is symmetrical across the vertical line through its maximum point.

The number $$\Tblue{c}$$ is the value of the function for $$x=0$$. It is the intercept with the $$y$$-axis.

Graphs of the two quadratic functions $$y = 2x^2-2x-2$$ (left) and $$y = - x^2 +4$$ (right) and a linear function $$y = 2x + 2$$ (centre).
Graphs of the two quadratic functions $$y = 2x^2-2x-2$$ (left) and $$y = - x^2 +4$$ (right) and a linear function $$y = 2x + 2$$ (centre).