From: Andrew Lorimer <andrew@lorimer.id.au>
Date: Wed, 5 Dec 2018 01:18:32 +0000 (+1100)
Subject: cubic/quartic graphs
X-Git-Tag: yr12~288^2~2
X-Git-Url: https://git.lorimer.id.au/notes.git/diff_plain/2efab049b6d20d02e29a2df8b837b01e79abfb0a?hp=68fd9bb96fb4d5f34ef9fb518bc81b28fb820669

cubic/quartic graphs
---

diff --git a/methods/polynomials.md b/methods/polynomials.md
index e81e0b0..c312123 100644
--- a/methods/polynomials.md
+++ b/methods/polynomials.md
@@ -21,3 +21,30 @@ $$y=mx+c, \quad {x \over a} + {y \over b}=1$$
 
 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 $x=b,c,d,e$