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.
2. Axis of Symmetry: The vertical line that passes through the vertex is called the axis of symmetry, given by the equation:
   $x = -\frac{b}{2a}$
3. Y-intercept: The point where the graph intersects the y-axis is given by the value $c$, or $(0, c)$.
4. Roots (X-intercepts): The points where the graph intersects the x-axis can be found using the quadratic formula:
   $x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}$
   The expression $b^{2} - 4ac$ is known as the discriminant and determines the nature of the roots.

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:
   $x = -\frac{-4}{2 \cdot 2} = \frac{4}{4} = 1$
3. Substitute $x = 1$ into $f(x)$ to find $y$:
   $f(1) = 2(1)^{2} - 4(1) + 1 = 2 - 4 + 1 = -1$
   So, the vertex is $(1, -1)$.
4. Find the roots using the quadratic formula:
   $x = \frac{-(-4) \pm \sqrt{(-4)^{2} - 4(2)(1)}}{2(2)}$
   $= \frac{4 \pm \sqrt{16 - 8}}{4}$
   $= \frac{4 \pm \sqrt{8}}{4}$
   $= \frac{4 \pm 2\sqrt{2}}{4}$
   $= 1 \pm \frac{\sqrt{2}}{2}$
   Thus, the roots are $1 + \frac{\sqrt{2}}{2}$ and $1 - \frac{\sqrt{2}}{2}$.

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 $f(x) = x^{2} - 4$?
• What is the axis of symmetry for the parabola represented by $f(x) = -x^{2} + 2x + 3$?

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

Keywords:

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