Quadratic Functions Cheat Sheet

Quadratic functions are second-degree polynomial functions, characterized by a parabolic graph. They are fundamental in algebra and have wide applications in physics and engineering.

Core Principles

• Definition: A quadratic function is of the form $f(x) = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants and $a \neq 0$.
• Graph: The graph of a quadratic function is a parabola.
• Vertex: The highest or lowest point on the parabola.
• Axis of Symmetry: A vertical line that divides the parabola into two mirror images.
• Roots/Zeros: The x-values where the function equals zero ($f(x) = 0$).
• Discriminant: Determines the nature of the roots.
• Standard Form: $f(x) = ax^2 + bx + c$.
• Vertex Form: $f(x) = a(x-h)^2 + k$, where $(h, k)$ is the vertex.
• Factored Form: $f(x) = a(x-r_1)(x-r_2)$, where $r_1$ and $r_2$ are the roots.

Action Steps

• Identify coefficients $a$, $b$, and $c$ from the standard form.
• Calculate the x-coordinate of the vertex using $h = -b / (2a)$.
• Calculate the y-coordinate of the vertex by substituting $h$ into the function: $k = f(h)$.
• Determine the axis of symmetry: $x = h$.
• Find the roots using the quadratic formula or factoring.
• Analyze the discriminant ($\Delta$) to understand the nature of the roots: $\Delta > 0$ (two distinct real roots), $\Delta = 0$ (one real root/repeated), $\Delta < 0$ (two complex roots).
• Determine the parabola's direction: opens upward if $a > 0$, downward if $a < 0$.
• Find the y-intercept by setting $x = 0$ (which is always $c$).

Formulas

• Standard Form: $f(x) = ax^2 + bx + c$
• Vertex Formula (x-coordinate): $h = -b / (2a)$
• Vertex Formula (y-coordinate): $k = f(h)$
• Quadratic Formula (for roots): $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
• Discriminant: $\Delta = b^2 - 4ac$
• Vertex Form: $f(x) = a(x-h)^2 + k$
• Factored Form: $f(x) = a(x-r_1)(x-r_2)$

Key Terms

• Parabola: The U-shaped graph of a quadratic function.
• Vertex: The minimum or maximum point of the parabola.
• Axis of Symmetry: The vertical line passing through the vertex, dividing the parabola symmetrically.
• Roots (Zeros): The x-values where the parabola intersects the x-axis.
• Discriminant: The part of the quadratic formula under the square root ($b^2 - 4ac$), indicating the number and type of roots.
• Y-intercept: The point where the parabola crosses the y-axis (occurs at $x=0$).

Real World Examples

• Projectile motion: The path of a thrown ball follows a parabolic trajectory.
• Optimization problems: Finding maximum profit or minimum cost in business scenarios.
• Bridge design: The shape of suspension bridge cables often approximates a parabola.
• Antenna dishes: Parabolic reflectors focus signals to a single point.