A quadratic function is a polynomial function of degree 2, typically expressed in the form $f(x) = ax^2 + bx + c$ where $a$, $b$, and $c$ are constants and $a \neq 0$. The graph of a quadratic function is a parabola that opens either upwards or downwards.
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The vertex of a quadratic function given by $f(x) = ax^2 + bx + c$ can be found using the formula $(-\frac{b}{2a}, f(-\frac{b}{2a}))$.
The axis of symmetry of the parabola is the vertical line $x = -\frac{b}{2a}$.
The direction in which the parabola opens is determined by the sign of coefficient $a$: it opens upwards if $a > 0$ and downwards if $a < 0$.
The roots (or zeros) of the quadratic function can be found using the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.
The discriminant, given by $b^2 - 4ac$, determines the nature of the roots: if it's positive, there are two distinct real roots; if zero, one real root; if negative, two complex roots.
Review Questions
How do you determine whether a parabola opens upwards or downwards?
What is the formula for finding the vertex of a quadratic function?
What does the discriminant tell you about the roots of a quadratic equation?