**Perfect squares:** $a^2 \pm 2ab + b^2 = (a \pm b^2)$
**Completing the square (monic):** $x^2+bx+c=(x+{b\over2})^2+c-{b^2\over4}$
**Completing the square (non-monic):** $ax^2+bx+c=a(x-{b\over2a})^2+c-{b^2\over4a}$
-**Quadratic formula:** $x={{-b\pm\sqrt{b^2-4ac}}\over2a}$ where $\Delta=b^2-4ac$
+**Quadratic formula:** $x={{-b\pm\sqrt{b^2-4ac}}\over2a}$ where $\Delta=b^2-4ac$ (if $\Delta$ is a perfect square, rational roots)
#### Cubics
**Difference of cubes:** $a^3 - b^3 = (a-b)(a^2 + ab + b^2)$
**Sum of cubes:** $a^3 + b^3 = (a+b)(a^2 - ab + b^2)$
**Perfect cubes:** $a^3 \pm 3a^2b + 3ab^2 \pm b^3 = (a \pm b)^3$
+
+## Linear and quadratic graphs
+
+$$y=mx+c, \quad {x \over a} + {y \over b}=1$$
+
+Parallel lines - $m_1 = m_2$
+Perpendicular lines - $m_1 \times m_2 = -1$
+
+
+## Cubic graphs
+
+$$y=a(x-b)^3 + c$$
+
+- $m=0$ at *stationary point of inflection*
+- in form $y=(x-a)^2(x-b)$, local max at $x=a$, local min at $x=b$
+- in form $y=a(x-b)(x-c)(x-d)$: $x$-intercepts at $b, c, d$
+
+
+## Quartic graphs
+
+$$y=ax^4$$
+
+$$=a(x-b)(x-c)(x-d)(x-e)$$
+
+$$=ax^4+cd^2 (c \ge 0)$$
+
+$$=ax^2(x-b)(x-c)$$
+
+$$=a(x-b)^2(x-c)^2$$
+
+$$=a(x-b)(x-c)^3$$
+
+where
+- $x$-intercepts at $x=b,c,d,e$