From b195e8d9f717a88d5ed46370d6400b40d460b24e Mon Sep 17 00:00:00 2001
From: Andrew Lorimer <andrew@lorimer.id.au>
Date: Sat, 2 Feb 2019 16:07:46 +1100
Subject: [PATCH] tidy up polynomials notes & add more to transformations notes

---
 methods/polynomials.md     | 67 +++++++++++++++++++++++++++-----------
 methods/transformations.md | 21 +++++++++++-
 2 files changed, 68 insertions(+), 20 deletions(-)

diff --git a/methods/polynomials.md b/methods/polynomials.md
index a8b56af..c3d38f3 100644
--- a/methods/polynomials.md
+++ b/methods/polynomials.md
@@ -17,10 +17,17 @@
 
 ## Linear and quadratic graphs
 
-$$y=mx+c, \quad {x \over a} + {y \over b}=1$$
+### Forms of linear equations
 
-Parallel lines - $m_1 = m_2$  
-Perpendicular lines - $m_1 \times m_2 = -1$
+$y=mx+c$ where $m$ is gradient and $c$ is $y$-intercept  
+${x \over a} + {y \over b}=1$ where $m$ is gradient and $(x_1, y_1)$ lies on the graph  
+$y-y_1 = m(x-x_1)$ where $(a,0)$ and $(0,b)$ are $x$- and $y$-intercepts
+
+## Line properties
+
+Parallel lines: $m_1 = m_2$  
+Perpendicular lines: $m_1 \times m_2 = -1$  
+Distance: $\vec{AB} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$
 
 
 ## Cubic graphs
@@ -34,30 +41,52 @@ $$y=a(x-b)^3 + c$$
 
 ## Quartic graphs
 
-$$y=ax^4$$
+### Forms of quadratic equations
+$y=ax^4$  
+$y=a(x-b)(x-c)(x-d)(x-e)$  
+$y=ax^4+cd^2 (c \ge 0)$  
+$y=ax^2(x-b)(x-c)$  
+$y=a(x-b)^2(x-c)^2$  
+$y=a(x-b)(x-c)^3$
 
-$$=a(x-b)(x-c)(x-d)(x-e)$$
+## Literal equations
 
-$$=ax^4+cd^2 (c \ge 0)$$
+Equations with multiple pronumerals. Solutions are expressed in terms of pronumerals (parameters)
 
-$$=ax^2(x-b)(x-c)$$
+## Simultaneous equations (linear)
 
-$$=a(x-b)^2(x-c)^2$$
+- **Unique solution** - lines intersect at point
+- **Infinitely many solutions** - lines are equal
+- **No solution** - lines are parallel
 
-$$=a(x-b)(x-c)^3$$
 
-where
-- $x$-intercepts at $x=b,c,d,e$
 
-## Literal equations
+### Solving $\begin{cases}px + qy = a \\ rx + sy = b\end{cases}$ for one, infinite and no solutions
 
-Equations with multiple pronumerals. Solutions are expressed in terms of pronumerals (parameters))
+where all coefficients are known except for one, and $a, b$ are known
 
-## Simultaneous equations (linear)
+1. Write as matrices: $\begin{bmatrix}p & q \\ r & s \end{bmatrix}
+  \begin{bmatrix} x \\ y \end{bmatrix}
+  =
+  \begin{bmatrix} a \\ b \end{bmatrix}$
+2. Find determinant of first matrix: $\Delta = ps-qr$
+3. Let $\Delta = 0$ for number of solutions $\ne 1$  
+   or let $\Delta \ne 0$ for one unique solution.
+4. Solve determinant equation to find variable  
+   - *--- for infinite/no solutions: ---*
+5. Substitute variable into both original equations
+6. Rearrange equations so that LHS of each is the same
+7. If $\text{RHS}(1) = \text{RHS}(2)$, lines are coincident (infinite solutions)  
+   If $\text{RHS}(1) \ne \text{RHS}(2)$, lines are parallel (no solutions)
 
-- **Unique solution** - lines intersect at point
-- **Infinitely many solutions** - lines are equal
-- **No solution** - lines are parallel
+Or use Matrix -> `det` on CAS.
+
+### Solving $\begin{cases}a_1 x + b_1 y + c_1 z = d_1 \\
+a_2 x + b_2 y + c_2 z = d_2 \\
+a_3 x + b_3 y + c_3 z = d_3\end{cases}$
 
-Solving in matrix form - use inverse $A^{-1}= {1 \over {ad-bc}}\begin{bmatrix}d & -b \\ -c & a\end{bmatrix}$. $A^{-1}$ exists for infinite solutions or no solutions ($ad-bc=0$), does not exist for unique solutions ($ad-bc \ne 0$).  
-Or use Matrix -> `det` on CAS.
\ No newline at end of file
+- Use elimination
+- Generate two new equations with only two variables
+- Rearrange & solve
+- Substitute one variable into another equation to find another variable
+- etc.
\ No newline at end of file
diff --git a/methods/transformations.md b/methods/transformations.md
index f335793..9a0b47b 100644
--- a/methods/transformations.md
+++ b/methods/transformations.md
@@ -5,4 +5,23 @@
 - $|a|$ is the dilation factor of $|a|$ units parallel to $y$-axis or from $x$-axis
 - if $a<0$, graph is reflected over $x$-axis
 - $k$ - translation of $k$ units parallel to $y$-axis or from $x$-axis
-- $h$ - translation of $h$ units parallel to $x$-axis or from $y$-axis
\ No newline at end of file
+- $h$ - translation of $h$ units parallel to $x$-axis or from $y$-axis
+
+## Translations
+
+For $y = f(x)$, these processes are equivalent:
+
+- applying the translation $(x, y) \rightarrow (x + h, y + k)$ to the graph of $y = f$(x)$
+- replacing $x$ with $x − h$ and $y$ with $y − k$ to obtain $y − k = f (x − h)$
+
+## Dilations
+
+For the graph of $y = f(x)$, there are two pairs of equivalent processes:
+
+1. - Dilating from $x$-axis: $(x, y) \rightarrow (x, by)$
+   - Replacing $y$ with $y \over b$ to obtain $y = b f(x)$
+
+2. - Dilating from $y$-axis: $(x, y) \rightarrow (ax, y)$
+   - Replacing $x$ with $x \over a$ to obtain $y = f({x \over a})$
+
+For graph of $y={1 \over x}$, horizontal & vertical dilations are equivalent (symmetrical). If $y={a \over x}$, graph is contracted rather than dilated.
\ No newline at end of file
-- 
2.43.2