The Master Transformation Box
Standard Shapes to Memorize:- $y = x^2$ (Parabola)- $y = \sqrt{x}$ (Half-Parabola on side)- $y = \ln(x)$ (Passes through $(1,0)$, asymptote at $x=0$)- $y = 1/x$ (Hyperbola)
Dimensional Analysis of a Graph
The Slope ($dy/dx$) always has units of $[Unit_y / Unit_x]$. The Area ($\int y dx$) always has units of $[Unit_y \times Unit_x]$. In Physics, if your $y$-axis is Force $[MLT^{-2}]$ and your $x$-axis is Displacement $[L]$, the area is Work $[ML^2T^{-2}]$. Always check this to ensure your graph represents the physical quantity you intend.
Variations: Even and Odd Functions
- Even Functions [$f(x) = f(-x)$]: Symmetric about the $y$-axis (e.g., $x^2, \cos x$).
- Odd Functions [$f(-x) = -f(x)$]: Symmetric about the Origin (e.g., $x^3, \sin x$).
Shortcuts & Mnemonics
- The "Inside-Opposite" Rule: Changes inside the parentheses ($x-h$) do the opposite of what you expect ($-h$ moves right). Changes outside do exactly what you expect.
- Mnemonic: "Rise over Run for the slope; Area is the product's scope."
Edge Cases
- Point Discontinuity: If a function like $y = (x^2-1)/(x-1)$ is simplified, it looks like a line, but there is a "hole" at $x=1$ because the original denominator cannot be zero.
- Cusp/Corner: Points where the graph is continuous but the derivative does not exist (e.g., at $x=0$ for $y = |x|$).
- Infinite Discontinuity: Where the graph shoots to $\pm \infty$ (Asymptotes).
