From: Andrew Lorimer <andrew@lorimer.id.au>
Date: Thu, 3 Jan 2019 02:56:05 +0000 (+1100)
Subject: Merge branch 'master' of ssh://charles/tank/andrew/school/notes
X-Git-Tag: yr12~288
X-Git-Url: https://git.lorimer.id.au/notes.git/diff_plain/3817bcdc33d972532ed8fabb75aed04c11421235?ds=inline;hp=-c

Merge branch 'master' of ssh://charles/tank/andrew/school/notes
---

3817bcdc33d972532ed8fabb75aed04c11421235
diff --combined methods/polynomials.md
index 0542afc,c312123..b80ea63
--- a/methods/polynomials.md
+++ b/methods/polynomials.md
@@@ -4,13 -4,47 +4,47 @@@
  
  #### Quadratics
  **Quadratics:** $x^2 + bx + c = (x+m)(x+n)$ where $mn=c$, $m+n=b$  
 -**Difference of squares:** $a^2 - b=^2 = (a - b)(a + b)$  
 +**Difference of squares:** $a^2 - b^2 = (a - b)(a + b)$  
  **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 $x=b,c,d,e$