spec / graphing.mdon commit [spec] add graphs for Euler's method and bisector theorem (ac940ff)
   1---
   2geometry: margin=2cm
   3columns: 2
   4graphics: yes
   5---
   6
   7# Graphing techniques
   8
   9## Reciprocal continuous functions
  10
  11If $y=f(x)$, the reciprocal function is:
  12
  13$$y={1 \over f(x)}$$
  14
  15As $\quad f(x) \rightarrow \pm \infty,\quad {1 \over f(x)} \rightarrow 0^\pm$ (vert asymptote at $f(x)=0$)
  16
  17<!-- As $\quad x \rightarrow  \pm \infty,\quad {-1 \over x}$ -->
  18
  19\includegraphics[width=0.25\textwidth]{./graphics/recip-parabola.png}
  20\includegraphics[width=0.25\textwidth]{./graphics/recip-sin-cos.png}
  21
  22- reciprocal functions are always on the same side of $x=0$
  23- if $y=f(x)$ has a local max|min at $x=1$, then $y={1 \over f(x)}$ has a local max|min at $x=a$
  24- point of inflection at $P(1,1)$
  25
  26## Locus of points
  27
  28- set of points that satisfy a given condition
  29- path traced by a point that moves according to a condition
  30- graph on CAS - **conics**
  31
  32### Circular loci
  33
  34point $P(x,y)$ has a constant distance $r$ from point $C(a,b)$ (centre)
  35
  36
  37$$PC  = r$$
  38
  39$$(x-a)^2 + (y-b)^2 = r^2$$
  40
  41
  42
  43### Linear loci
  44
  45$$QP  =  RP $$
  46$$\sqrt{(x_Q-q_P)^2+(y_Q-y_P)^2} = \sqrt{(x_R-x_P)^2+(y_R-y_P)^2}$$
  47
  48points $Q$ and $R$ are fixed and have a perpendicular bisector $QR$. Therefore, any point on line $y=mx+c$ is equidistant from $QP$ and $RP$.
  49
  50Since the bisector of the line joining points $Q$ and $R$ is perpendicular to $QR $:
  51
  52$$m( QR ) \times m( RP ) = -1$$
  53
  54### Parabolic loci
  55
  56$$PD  =  PF $$
  57$$|y-z|=\sqrt{(x-x_F)^2+(y-y_F)^2}$$
  58$$(y-z)^2=(x-x_F)^2+(y-y_F)^2$$
  59
  60Distance of point $P(x,y)$ from fixed point $F(a,b)$ is equal to the distance of $P$ from $y=z \perp$.
  61
  62Fixed point $F$ is the **focus** (halfway between $y=z$ and $y=y_P$)
  63
  64Fixed line $x=z$ is the **directrix**
  65
  66### Elliptical loci
  67
  68Point $P$ moves so that the sum of its distances from two fixed points $F_1$ and $F_2$ is a constant $k$.
  69
  70$${F_1 P}  +   F_2 P  =k$$
  71
  72**Two** foci at $F_1$ and $F_2$
  73
  74Cartesian equation for ellipses:
  75$${(x-h)^2 \over a^2} + {(y-k)^2 \over b^2} = 1$$
  76centered at $(h,k)$. Width is $2a$, height is $2b$.
  77
  78### Transformations
  79$$(x,y) \rightarrow (x \prime, y \prime)$$
  80
  81where $x \prime$ and $y \prime$ are the transformation factors (dilation away from $x$-axis means coefficient of $y$ increases in $y \prime$, and vice versa).
  82
  83Transformed equation is the same as initial equation with each term divided by its dilation coefficients (must be in terms of $x\prime$ and $y\prime$).
  84
  85e.g.
  86
  87$x^2 + y^2 = 1$ is dilated $3$ from $x$, $5$ from $y$.
  88Transformation rule is $(x\prime,y\prime) = (5x,3y)$
  89$x={x\prime \over 5},\quad y={y\prime \over 3}$
  90
  91Equation $x^2 + y^2=1$ becomes
  92
  93$${(x\prime)^2 \over 25}+ {(y\prime)^2 \over 9}=1$$
  94
  95
  96### Hyperbolic loci
  97
  98$$|(F_2P - F_1P  )| = k$$
  99
 100Cartesian equation for hyperbolas ($a$ and $b$ are dilation factors):
 101$${(x-h)^2 \over a^2} - {(y-k)^2 \over b^2} = 1$$
 102
 103Distance between vertices is $2a$
 104Vertices given by $(h \pm a, k)$
 105
 106Asymptotes at $y=\pm {b \over a}(x-h)+k$
 107To make hyperbola up/down rather than left/right, swap $x$ and $y$
 108
 109$y^2-x^2=1$ produces hyperbola shifted 90 $^\circ$ (top and bottom of asymptotes)
 110
 111## Parametric equations
 112
 113Parametric curve:
 114
 115$$x=f(t), \quad y=g(t)$$
 116
 117$t$ is the parameter
 118
 119To convert to cartesian, solve like simultaneous equations
 120
 121## Polar coordinates
 122
 123$$x = r\cos\theta, \quad y = r\sin\theta$$
 124
 125### Spirals
 126$$r={\theta \over n\pi}$$
 127- solve intercepts for multiples of $\pi \over 2$
 128- or draw table of values for $r$ and $\theta$ for each $n\pi \over 2$
 129
 130### Circles
 131$$r=a$$
 132
 133### Lines
 134
 135Horizontal: $r={n \over \sin \theta}$
 136Vertical: $r={n \over \cos \theta}$
 137
 138### Cardioids
 139
 140$$r=a(n+ \cos\theta)$$
 141
 142### Roses
 143
 144$$r=\cos(k\theta)$$
 145
 146If $k$ is odd, half of the petals will overlap (hence there are $n$ petals)
 147
 148If $k$ is even, petals will not overlap (hence $2n$ petals)
 149
 150\includegraphics[width=0.5\textwidth]{./graphics/rose.png}
 151
 152
 153### Solving polar graphs
 154
 155solve in terms of $r$
 156
 157e.g. $x=4$
 158
 159$r\cos\theta = 4$
 160
 161$r={4 \over \cos\theta}$
 162
 163
 164---
 165
 166e.g. $y=x^2$
 167
 168$r\sin\theta = r^2 \cos^2 \theta$
 169
 170$\sin \theta = r \cos^2 \theta$
 171
 172$r = {\sin \theta \over \cos^2\theta} = \tan\theta \sec\theta$
 173
 174---
 175
 176e.g. $r=6\cos \theta\quad$ *(multiply by $r$)*
 177
 178$r^2=6r\cos\theta$
 179
 180$x^2+y^2=6x$
 181
 182complete the square