From b195e8d9f717a88d5ed46370d6400b40d460b24e Mon Sep 17 00:00:00 2001 From: Andrew Lorimer 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