Properties of Quadratics

Quadratic functions are polynomials of degree 2 and can be expressed in the standard 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.

Key Properties:

1. Vertex: The vertex of the parabola is the point where it changes direction. It can be found using the formula:
   $x = -\frac{b}{2a}$
   The corresponding $y$-value can be found by substituting this $x$ back into the function.

Example:

Find the vertex and roots of the quadratic function $f(x) = 2x^{2} - 4x + 1$.

1. Identify coefficients:
   $a = 2$, $b = -4$, $c = 1$
2. Find the vertex:
   $x = -\frac{-4}{2 \cdot 2} = \frac{4}{4} = 1$

Key Questions:

• What is the vertex of the quadratic $f(x) = 3x^{2} + 6x + 2$?
• How do you determine the number of roots for the function $f(x) = x^{2} - 4$?

• Find the y-intercept of the quadratic function $f(x) = 5x^{2} - 3x + 7$.

Keywords:

• Quadratic function
• Vertex
• Axis of symmetry
• Roots
• Discriminant