Analysis of quadratic trinomials

Analyzing a quadratic trinomial $ax^2 + bx + c$ involves:

• Determining the direction of the parabola (upward if $a > 0$, downward if $a < 0$)
• Finding the roots (real or complex) using the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
• Locating the vertex at $\left(-\frac{b}{2a}, \frac{4ac-b^2}{4a}\right)$
• Identifying the axis of symmetry $x = -\frac{b}{2a}$
• Calculating the $y$-intercept $c$

Important properties

• The discriminant $D = b^2 - 4ac$ tells us the number and type of roots.

• The vertex gives the minimum (if $a > 0$) or maximum (if $a < 0$) value.

• The axis of symmetry divides the parabola into two mirror images.

• The sign of $a$ determines if the parabola opens up or down.