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

$f(x) = \Tred{-2} x^2 + \Tblue{1}$ and $g(x) = \Tred{2}x^2 \Tgreen{-3} x - \Tblue{6}$ 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 $f(0)$. 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).